Differential Equations

Problem 1

Find the equation for water level $L$ receding at 0.5 ft/day when initial level is 34 ft. Determine how many days until level reaches 26 ft.

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Problem 2

Find the time it takes for a tranquilizer with a half-life of 50 hours to decay to $95\%$ of its original dosage using the exponential decay model $A=A_{0} e^{k t}$ .

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Problem 3

Fetal head circumference $H$ depends on age $t$ in weeks as $H=-30.04+1.802t^2-0.9032t^2\log t$ . (a) Calculate $\frac{dH}{dt}$ . (b) Is $\frac{dH}{dt}$ larger at $t=8$ or $t=36$ weeks? (c) Repeat (b) for $\frac{1}{H}\frac{dH}{dt}$ . (a) $\frac{dH}{dt}=3.604t-0.9032t\log t-0.9032t$ .

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Problem 4

Find the value of $y_0(2)$ where $y_0$ is the solution to the ODE $xy' + 2y = 2x$ with $y(1) = 3$ .

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Problem 5

Find the amount of radioactive substance present (in grams) after 2 years, given the formula $y=8,000(2)^{-0.3 t}$ , where $t$ is in months. Round the result to three decimal places if needed.

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Problem 6

Classify the equilibrium at the origin in $X = CX$ where $C = \begin{pmatrix} 9 & -11 \\ -11 & 9 \end{pmatrix}$ as a sink, saddle, or source.

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Problem 7

Trouver la forme de la solution particulière $y_p$ d'une équation différentielle avec $r(x) = 4x e^{3x} \cos(2x)$ et solutions homogènes $y_h = C_1 e^{3x} + C_2 \cos(2x) + C_3 \sin(2x)$ .

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Problem 8

Solve the ordinary differential equation $y-3x + (4y+3x)\frac{dy}{dx}=0$ using the homogeneous method.

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Problem 9

Find all solutions to the second-order differential equation $y^{\prime \prime}=x \cos(x)$ .

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Problem 10

Find the rat population in 2000 given the formula $n(t)=86 e^{0.04 t}$ where $t$ is years since 2000 and $n(t)$ is in millions.

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Problem 11

Insect population of $\mathrm{P}(t) = 600e^{0.02t}$ at time $t$ days. Find: (a) population at $t=0$ , (b) growth rate, (c) graph, (d) population after 10 days, (e) when population reaches 900, (f) when population doubles.

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Problem 12

Solve the differential equation $\frac{dy}{dx} = \frac{1}{5} \sqrt{y} \cos^2 \sqrt{y}$ .

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Problem 13

Solve for t in the equation $h = 7 + 35t - 16t^2$ , given that $h = 25$ .

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Problem 14

Find the solution to the ODE $y' = -1.7xy$ with $y(0) = 9.6$ , and evaluate the solution at $x = 1.3$ .

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Problem 15

Find $\Delta T$ in the equation $q = m \times \Delta T \times C$ , where $q$ , $m$ , and $C$ are known.

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Problem 16

Solve for the general solution of the differential equation $(x^2-xy+y^2)dx-xy\,dx=0$ . If homogeneous, use $y=vx$ .

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Problem 17

Erbium isotope with 12-hour half-life, initial 22g. Find: a) $A(t)$ function, b) decay rate $A'(t)$ , c) decay rate at 11 hours.
a) $A(t) = 22e^{-0.0577t}$ grams b) $A'(t) = -1.2694e^{-0.0577t}$ grams/hour c) $A'(11) = 0.2069$ grams/hour

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Problem 18

Find the integrating factor for the first-order linear ODE $y' = -y + \cos x$ .

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Problem 19

Find the general solution for the differential equation $\frac{d A}{d t}=-7 A$ , where $A(t)=Ce^{-7t}$ .

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Problem 20

Solve the ODE $(6+t) \frac{d u}{d t} + u = 6+t$ for $t > 0$ .

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Problem 21

Find the limit of the solution to the differential equation $y' = y^3 - 4y$ with initial condition $y(0) = 0.2$ as $t$ approaches infinity.

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Problem 22

Use Euler's method with step size 0.1 to find $y(0.2)$ , where $y(x)$ satisfies $y'=2x+3y^2, y(0)=0$ .

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Problem 23

Find $\frac{dx}{dt}$ when $x=-2$ and $y=2$ given $y^2 + xy - 3x = 6$ and $\frac{dy}{dt} = -2$ .

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Problem 24

(a) Find depth where light intensity is half of surface intensity $L$ . Depth = $D$ meters. (b) Light intensity at 11 m depth = $I = L \cdot e^{(-1.95 \cdot 11)}$ . (c) Find depth where light intensity is 2.4% of surface intensity $L$ . Depth = $D$ meters.

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Problem 25

Find the general solution of the second-order linear ODE $y'' + y = 0$ .

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Problem 26

Find the solution of the initial value problem $g'(x)=8x(x^7-1/8); g(1)=1$ . The solution is $g(x)=\square$ .

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Problem 27

Solve the ordinary differential equation $y'-2y=0$ .

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Problem 28

1. a. Solve $2^{x+1}=5^{2x-1}$ for $x$ , exact or rounded to 3 decimals. b. Solve $\log(x^2-10)+\log(9)=1$ for $x$ , give exact (g) and check (s). c. The amount $A$ of a radioactive substance decays over time $t$ (years) as $A(t)=A_0\cdot(1/2)^{t/k}$ , where $k$ is the half-life. Given $A(3)=96.2\%A_0$ , compute the half-life rounded to full years.

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Problem 29

Cooling liquid temperature model: $T(t) = 80(\frac{1}{2})^{\frac{t}{10}} + 20$ . Sketch graph, find temp. after 40 mins, time to cool to 45 degrees.

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Problem 30

Find the initial number of people ill with the flu, given the function $N(t)=\frac{5000}{1+999 e^{-t}}$ models the number of people ill $t$ weeks after the outbreak. Round the result to the nearest person.

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Problem 31

Find an exponential model $P=a\cdot b^t$ for a population of 16,000 organisms growing by $6.9\%$ annually.

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Problem 32

Solve the initial value problem $\frac{dv}{dt}=\frac{2}{t\sqrt{t^2-1}}, t>1, v(2)=0$ and find $v(t)$ .

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Problem 33

Find the height of an aircraft with atmospheric pressure $p=259$ mm Hg using the formula $p=760 e^{-0.145 \mathrm{h}}$ . Round the height to two decimal places.

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Problem 34

Find the value of a quantity that decays exponentially at $8.5\%$ every 9 weeks, starting at 390, after 42 days, to the nearest hundredth.

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Problem 35

Sketch direction field and solutions for differential equation $\frac{dy}{dx} = y^2 - x$ using isoclines for slopes $m = -1, 0, 1, 2$ .

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Problem 36

Find the general solution for the third-order linear ODE $y'''- y''- y' + y = g(t)$ .

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Problem 37

The two airplanes will be at the same altitude in $\sqrt{\frac{46400}{3200 + 2600}}$ minutes, when their altitude will be $23200$ feet.

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Problem 38

A lake covers 11 square km, decreasing exponentially by 2% per year: $A(t)=11 \cdot(0.98)^{t}$ . By what factor does the area decrease in 10 years?

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Problem 39

Find the solutions to $x-x e^{5 x+2}=0$ . Then find the second derivative of $f(x)$ at $x=1$ , given $f'(x)+5f(x^2)=f^2(x)$ and $f(1)=2$ .

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Problem 40

Find the largest interval $I$ over which the solution $y(x) = \frac{1}{x^2 + C}$ to the ODE $y' + 2x^6 = 0$ is defined, given $y(4) = \frac{1}{7}$ .

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Problem 41

Find the carrying capacity of a population that grows according to the logistic equation $\frac{dP}{dt} = 0.084P - 0.00168P^2$ , where $t$ is measured in weeks.

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Problem 42

Exposure time to reach coma-inducing CO level: $T=.0002 x^{2}-.316 x+127.9$ , where $x$ is CO in ppm. Find exposure time when $x=560$ ppm. Round to nearest tenth.

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Problem 43

Solve the separable differential equation $\frac{dy}{dx} + 4xy^2 = 0$ .

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Problem 44

Solve the differential equation $y' = \frac{\sin \sqrt{x}}{7 \sqrt{x}}$ .

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Problem 45

Solve the differential equation $\frac{d x}{d t}=5 t^{-1}+5$ for the given initial conditions.

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Problem 46

Solve the initial value problem: $\frac{d y}{d x}=8 x e^{-6 y}, \quad y(0)=0$ .

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Problem 47

Identify the exact ordinary differential equation (ODE) from the given options.
$(x+y) dx + (y+x) dy = 0$ $(xy+y) dx + (xy+x) dy = 0$ $(y+2) dx + (2x+1) dy = 0$

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Problem 48

Solve the differential equation $3 y y' = 5 x$ .

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Problem 49

Find $T_2$ given $t_2 \cdot \frac{P_1 V_1}{T_1} = \frac{P_2 V_2}{T_2}$ .

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Problem 50

Find the solution to the differential equation $\frac{d y}{d x}=e^{2 x}+2 y$ . Options: A. $y=e^{2 x}$ , B. $y=-e^{2 x}$ , C. $y=x e^{2 x}$ , D. $y=-x e^{2 x}$ .

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Problem 51

Radioactive fallout from a 1981 nuclear lab explosion is described by $f(x) = 1000(0.5)^{\frac{x}{30}}$ , where $f(x)$ is the remaining amount (kg) $x$ years later. Determine $f(40)$ and if the area will be safe by 2021.

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Problem 52

Find the Laplace transform of $3te^{5t}$ using a table of Laplace transforms. Integration by parts may be useful.

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Problem 53

26) Find the integral for the volume of a solid with base $R$ and height 5 times the base: (a) $25 \int_{a}^{b}(g(x)-f(x))^{2} dx$ (b) $5 \int_{a}^{b}(g(x)-f(x)) dx$ (c) $5 \int_{a}^{b}(f(x)-g(x)) dx$ (d) $5 \int_{a}^{b}(f(x)-g(x))^{2} dx$
27) Solve $\frac{dy}{dx}=\frac{x}{y}$ with $y=4$ at $x=2$ : (a) $\sqrt{\frac{1}{2} x^{2}+14}$ (b) $\sqrt{2 x^{2}+8}$ (c) $\sqrt{x^{2}+6}$ (d) $\sqrt{x^{2}+12}$

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Problem 54

Solve the differential equation: $x y \, dx - (x+2) \, dy = 0$ .

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Problem 55

Un point $\mathrm{P}$ de masse $\mathrm{m}$ se déplace en coordonnées polaires. Trouvez les équations du mouvement et les expressions pour $\mathrm{r}(t)$ et $\omega$ quand $\omega$ est constant.

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Problem 56

Find the function $f(x)$ given that $f^{\prime}(x)=2 f(x)$ and $f(2)=1$ .

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Problem 57

Analyze the stability of equilibrium points for $\ddot{x}=(x-a)(x^{2}-a)$ across all real values of $a$ .

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Problem 58

Solve the differential equation: $y^{\prime \prime}-7 y^{\prime}+12 y=0$ .

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Problem 59

Find the salt concentration in a 100-gallon tank after 30 mins, starting with 10 lbs salt and adding brine at $4 \mathrm{gal/min}$ .

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Problem 60

13. Solve $\left[x \sin ^{2}\left(\frac{y}{x}\right)-y\right] dx + x dy = 0$ for $y=\frac{\pi}{4}$ at $x=1$ .
14. Find $\frac{dy}{dx}-\frac{y}{x}+\operatorname{cosec}\left(\frac{y}{x}\right)=0$ for $y=0$ at $x=1$ .

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Problem 61

Upton Chuck free falls for 2.60 seconds. Calculate his final velocity and the distance fallen. Use $v = gt$ and $d = \frac{1}{2}gt^2$ .

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Problem 62

Identify type, order, and degree of these differential equations:
1. $\left(\frac{d^{2} w}{d x^{2}}\right)^{3}+x y \frac{d w}{d x}+w=k$
2. $\frac{d^{3} s}{d t^{3}}-k^{2} t^{2}+a^{4}=0$
3. $y^{\prime \prime}+\left(y^{\prime}\right)^{2}=y$
4. $7\left(\frac{\partial x}{\partial y}\right)^{4}-\left(\frac{\partial x}{\partial y}\right)^{2}+c=\frac{\partial^{2} x}{\partial y^{2}}$
5. $\left(x^{2}+y^{2}\right) d x=2 x^{2} y^{2} d y$
6. $\left(\frac{d^{3} s}{d t^{3}}\right)^{2}+s\left(\frac{d^{2} s}{d t^{2}}\right)^{3}+2 s t=0$
7. $2 x^{3}-8=\frac{d^{4} y}{d x^{4}}$
8. $x^{2} d x+y^{2} d y=\frac{d^{2} y}{d x^{2}}$
9. $\frac{\partial u}{\partial t}=h^{2}\left(\frac{\partial^{2} u}{\partial x^{2}}+\frac{\partial^{2} u}{\partial u^{2}}\right)$
10. $\left(\frac{\partial u}{\partial v}\right)^{3}+x^{2}\left(\frac{\partial^{2} u}{\partial x}\right)+v^{3}\left(\frac{\partial u}{\partial v}\right)^{3}=\frac{d v}{x}$

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Problem 63

Solve the DE: $(1-x) y' = y^2$ and determine if the solution is general or particular. Also solve: $a^2 dx = y \sqrt{y^2 - a^2} dy$ .

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Problem 64

Solve the differential equation $y' = x e^{-y - x^2}$ . Is the solution general or particular?

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Problem 65

Solve the separable differential equation $x y^{3} dx + (y+1)e^{-x} dy = 0$ and identify the type of solution. Also, solve $y' = x y^{3}$ .

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Problem 66

Solve these differential equations and identify solutions as general or particular:
1. $3v \frac{dv}{dy}=9$ , initial: $x=x_{0}$ , $v=v_{0}$ .
2. $y'=xy^{3}$ .
3. $\frac{dr}{dt}=-2rt$ , initial: $t=0$ , $r=r_{0}$ .
4. $\frac{dV}{dP}=\frac{-V}{P}$ .
5. $r(1+\ln x)dx+(1+\ln y)dy=0$ .
6. $\frac{dy}{dt}=\sin^{2}x\cos^{3}E$ .

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Problem 67

Solve the differential equation: $\frac{dy}{dx} = 3y$ .

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Problem 68

Solve the differential equation $\frac{d y}{d x}=\frac{3 y-7 x+7}{3 x-7 y-3}$ .

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Problem 69

Solve the differential equation: $y' - \frac{2x-1}{x^2-x} y = 1$ .

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Problem 70

Solve the differential equation: $y' - \frac{x^2 x - 1}{x^2 - x} y = 1$ .

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Problem 71

A ball is dropped from a 94 ft building. Find the time to fall half the distance and to ground level: (a) $t=$ sec, (b) $t=$ sec.

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Problem 72

A rumor spreads in a school. Given $y^{\prime}(t)=k y(1-y)$ , find when $90\%$ of 1000 students hear it.

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Problem 73

Solve the logistic equation $\frac{d y}{d x}=5 y\left(1-\frac{y}{10}\right)$ for $y$ and find $y(0)=6$ .

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Problem 74

Find the function $y(x)=7+\int_{2}^{x} \sin(t^{2}) dt$ given $\frac{d y}{d x}=\sin(x^{2})$ and $y(2)=7$ .

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Problem 75

Find the exact value of $y(2)$ given the differential equation $\frac{d y}{d x}=3 x^{2}+2$ and $y(1)=6$ .

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Problem 76

Solve the equation $\frac{d y}{d x}=-4(1+y^{2}) x^{7}$ using separation of variables. Let $C$ be the constant. $y=$

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Problem 77

Solve the equation $(10+x^{4}) \frac{dy}{dx}=\frac{x^{3}}{y}$ using separation of variables. Find $y^{2}=$ .

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Problem 78

Solve the equation $e^{-y} \sin (x)+\frac{d y}{d x}=0$ using separation of variables. Find $y=C$ .

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Problem 79

Solve the initial value problem using separation of variables: $\frac{d y}{d x}-x^{2} e^{y}=9 e^{y}$ , $y(0)=3$ . Find $y=$

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Problem 80

Solve the equation $\frac{d y}{d t}=5 y$ with the initial condition $y(2)=2$ . Find $y=$

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Problem 81

Solve the equation $\frac{d u}{d t}=e^{3 u+4 t}$ with $u(0)=13$ . Find $u(t)$ .

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Problem 82

Solve the equation $\frac{d y}{d x}=\frac{-0.4}{\cos (y)}$ with initial condition $y(0)=\frac{\pi}{3}$ . Find $y(x)$ .

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Problem 83

Solve the differential equation $y' = x^4 \tan(y)$ for the general solution, using the constant $\mathrm{C}$ .

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Problem 84

A rocket starts from rest with mass $m_0$ and burns fuel at rate $k$ . Find $v(t)$ from $m = m_0 - kt$ and $m \frac{dv}{dt} = ck - mg$ .
(a) $v(t) = \, \mathrm{m/sec}$
(b) If fuel is 80% of $m_0$ and lasts 110s, find $v(110)$ with $g=9.8 \, \mathrm{m/s}^2$ and $c=2500 \, \mathrm{m/s}$ .
$v(110) = \, \mathrm{m/sec}$ [Round to nearest whole number]

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Problem 85

Solve the initial value problem: $y'' + 18x = 0$ , with $y(0) = 2$ and $y'(0) = 2$ . Find $y = \quad: \cdots \quad:$

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Problem 86

Solve the initial value problem: $y'' + 18x = 0$ , with $y(0) = 2$ and $y'(0) = 2$ . Find $y =$ .

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Problem 87

Solve the initial value problem: $y^{\prime \prime}+18 x=0$ , with $y(0)=2$ , $y^{\prime}(0)=2$ . Find $y=...$

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Problem 88

Find all values of $r$ for which $y=r x^{2}$ satisfies the equation $y^{\prime}=9 x$ . Enter answers as a list. $r=$

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Problem 89

Find the growth constant $k$ for a trout population growing from 2500 to 6250 in 1 year, with a capacity of 25000. When will it reach 12900? $k=$ $\therefore \mathrm{yr}^{-1}$ Time to 12900: $\approx$ - years.

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Problem 90

A rocket starts from rest with mass $m_0$ and burns fuel at rate $k$ . Find $v(t)$ from $m \frac{d v}{d t}=c k-m g$ .
(a) $v(t)=\quad \theta_{-} \mathrm{m} / \mathrm{sec}$
(b) If fuel is 80% of $m_0$ and lasts 110 s, find $v(110)$ with $g=9.8 \mathrm{~m/s}^2$ and $c=2500 \mathrm{~m/s}$ . Round to nearest whole number.

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Problem 91

False. The solutions to the differential equation $\frac{d y}{d x}=y^{2}-4 y$ include other values.

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Problem 92

Find the value of $K$ in the solution $y(t)=\frac{K}{1+A e^{-r t}}$ for the IVP $\frac{d y}{d t}=3 y(1-\frac{y}{12})$ , $y(0)=2$ .

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Problem 93

Solve the initial value problem: $\frac{d y}{d t}=3 y\left(1-\frac{y}{12}\right)$ , $y(0)=2$ . Find $y(t)$ .

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Problem 94

Find the value of $y(1)$ for the initial value problem $\frac{d y}{d x}=\frac{\ln (x)}{x}$ with $y(e)=3$ .

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Problem 95

Given that $f(x)$ is said to be nonnegative but has negative values in the table, clarify this. Find $y(4)$ from $\frac{d y}{d x}=y^{2} f^{\prime}(x), \quad y(3)=1$ .

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Problem 96

How long (in yr) will it take for 1.00 g of Strontium-90 to decay to 0.200 g, given a half-life of 28.1 yr?

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Problem 97

Solve the initial value problem $y' = 9ty^2$ , $y(0) = y_0$ , and find how the solution interval depends on $y_0$ .

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Problem 98

Solve the initial value problem $y^{\prime}+7 y=g(t)$ with $y(0)=0$ , where $g(t)=1$ for $0 \leq t \leq 1$ and $g(t)=0$ for $t>1$ .

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Problem 99

A coffee cup has $140 \mathrm{mg}$ caffeine. If it decreases by $10 \%$ hourly, how long to eliminate half?

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Problem 100

Given 800g of strontium-90 decaying at $-2.44\%$ , find: (a) decay rate, (b) amount left after 40 years, (c) time for 200g left, (d) half-life. Use $N = N_0 e^{-\lambda t}$ .

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1 2 3 4 5 6 7 8 9

Differential Equations

Problem 1

Find the equation for water level LLL receding at 0.5 ft/day when initial level is 34 ft. Determine how many days until level reaches 26 ft.

Problem 2

Find the time it takes for a tranquilizer with a half-life of 50 hours to decay to 95%95\%95% of its original dosage using the exponential decay model A=A0ektA=A_{0} e^{k t}A=A0​ekt.

Problem 3

Problem 4

Find the value of y0(2)y_0(2)y0​(2) where y0y_0y0​ is the solution to the ODE xy′+2y=2xxy' + 2y = 2xxy′+2y=2x with y(1)=3y(1) = 3y(1)=3.

Problem 5

Find the amount of radioactive substance present (in grams) after 2 years, given the formula y=8,000(2)−0.3ty=8,000(2)^{-0.3 t}y=8,000(2)−0.3t, where ttt is in months. Round the result to three decimal places if needed.

Problem 6

Classify the equilibrium at the origin in X=CXX = CXX=CX where C=(9−11−119)C = \begin{pmatrix} 9 & -11 \\ -11 & 9 \end{pmatrix}C=(9−11​−119​) as a sink, saddle, or source.

Problem 7

Problem 8

Solve the ordinary differential equation y−3x+(4y+3x)dydx=0y-3x + (4y+3x)\frac{dy}{dx}=0y−3x+(4y+3x)dxdy​=0 using the homogeneous method.

Problem 9

Find all solutions to the second-order differential equation y′′=xcos⁡(x)y^{\prime \prime}=x \cos(x)y′′=xcos(x).

Problem 10

Find the rat population in 2000 given the formula n(t)=86e0.04tn(t)=86 e^{0.04 t}n(t)=86e0.04t where ttt is years since 2000 and n(t)n(t)n(t) is in millions.

Problem 11

Insect population of P(t)=600e0.02t\mathrm{P}(t) = 600e^{0.02t}P(t)=600e0.02t at time ttt days. Find: (a) population at t=0t=0t=0, (b) growth rate, (c) graph, (d) population after 10 days, (e) when population reaches 900, (f) when population doubles.

Problem 12

Solve the differential equation dydx=15ycos⁡2y\frac{dy}{dx} = \frac{1}{5} \sqrt{y} \cos^2 \sqrt{y}dxdy​=51​y​cos2y​.

Problem 13

Solve for t in the equation h=7+35t−16t2h = 7 + 35t - 16t^2h=7+35t−16t2, given that h=25h = 25h=25.

Problem 14

Find the solution to the ODE y′=−1.7xyy' = -1.7xyy′=−1.7xy with y(0)=9.6y(0) = 9.6y(0)=9.6, and evaluate the solution at x=1.3x = 1.3x=1.3.

Problem 15

Find ΔT\Delta TΔT in the equation q=m×ΔT×Cq = m \times \Delta T \times Cq=m×ΔT×C, where qqq, mmm, and CCC are known.

Problem 16

Solve for the general solution of the differential equation (x2−xy+y2)dx−xy dx=0(x^2-xy+y^2)dx-xy\,dx=0(x2−xy+y2)dx−xydx=0. If homogeneous, use y=vxy=vxy=vx.

Problem 17

Problem 18

Find the integrating factor for the first-order linear ODE y′=−y+cos⁡xy' = -y + \cos xy′=−y+cosx.

Problem 19

Find the general solution for the differential equation dAdt=−7A\frac{d A}{d t}=-7 AdtdA​=−7A, where A(t)=Ce−7tA(t)=Ce^{-7t}A(t)=Ce−7t.

Problem 20

Solve the ODE (6+t)dudt+u=6+t(6+t) \frac{d u}{d t} + u = 6+t(6+t)dtdu​+u=6+t for t>0t > 0t>0.

Problem 21

Find the limit of the solution to the differential equation y′=y3−4yy' = y^3 - 4yy′=y3−4y with initial condition y(0)=0.2y(0) = 0.2y(0)=0.2 as ttt approaches infinity.

Problem 22

Use Euler's method with step size 0.1 to find y(0.2)y(0.2)y(0.2), where y(x)y(x)y(x) satisfies y′=2x+3y2,y(0)=0y'=2x+3y^2, y(0)=0y′=2x+3y2,y(0)=0.

Problem 23

Find dxdt\frac{dx}{dt}dtdx​ when x=−2x=-2x=−2 and y=2y=2y=2 given y2+xy−3x=6y^2 + xy - 3x = 6y2+xy−3x=6 and dydt=−2\frac{dy}{dt} = -2dtdy​=−2.

Problem 24

(a) Find depth where light intensity is half of surface intensity LLL. Depth = DDD meters. (b) Light intensity at 11 m depth = I=L⋅e(−1.95⋅11)I = L \cdot e^{(-1.95 \cdot 11)}I=L⋅e(−1.95⋅11). (c) Find depth where light intensity is 2.4% of surface intensity LLL. Depth = DDD meters.

Problem 25

Find the general solution of the second-order linear ODE y′′+y=0y'' + y = 0y′′+y=0.

Problem 26

Find the solution of the initial value problem g′(x)=8x(x7−1/8);g(1)=1g'(x)=8x(x^7-1/8); g(1)=1g′(x)=8x(x7−1/8);g(1)=1. The solution is g(x)=□g(x)=\squareg(x)=□.

Problem 27

Solve the ordinary differential equation y′−2y=0y'-2y=0y′−2y=0.

Problem 28

Problem 29

Cooling liquid temperature model: T(t)=80(12)t10+20T(t) = 80(\frac{1}{2})^{\frac{t}{10}} + 20T(t)=80(21​)10t​+20. Sketch graph, find temp. after 40 mins, time to cool to 45 degrees.

Problem 30

Find the initial number of people ill with the flu, given the function N(t)=50001+999e−tN(t)=\frac{5000}{1+999 e^{-t}}N(t)=1+999e−t5000​ models the number of people ill ttt weeks after the outbreak. Round the result to the nearest person.

Problem 31

Find an exponential model P=a⋅btP=a\cdot b^tP=a⋅bt for a population of 16,000 organisms growing by 6.9%6.9\%6.9% annually.

Problem 32

Solve the initial value problem dvdt=2tt2−1,t>1,v(2)=0\frac{dv}{dt}=\frac{2}{t\sqrt{t^2-1}}, t>1, v(2)=0dtdv​=tt2−1​2​,t>1,v(2)=0 and find v(t)v(t)v(t).

Problem 33

Find the height of an aircraft with atmospheric pressure p=259p=259p=259 mm Hg using the formula p=760e−0.145hp=760 e^{-0.145 \mathrm{h}}p=760e−0.145h. Round the height to two decimal places.

Problem 34

Find the value of a quantity that decays exponentially at 8.5%8.5\%8.5% every 9 weeks, starting at 390, after 42 days, to the nearest hundredth.

Problem 35

Sketch direction field and solutions for differential equation dydx=y2−x\frac{dy}{dx} = y^2 - xdxdy​=y2−x using isoclines for slopes m=−1,0,1,2m = -1, 0, 1, 2m=−1,0,1,2.

Problem 36

Find the general solution for the third-order linear ODE y′′′−y′′−y′+y=g(t)y'''- y''- y' + y = g(t)y′′′−y′′−y′+y=g(t).

Problem 37

The two airplanes will be at the same altitude in 464003200+2600\sqrt{\frac{46400}{3200 + 2600}}3200+260046400​​ minutes, when their altitude will be 232002320023200 feet.

Problem 38

A lake covers 11 square km, decreasing exponentially by 2% per year: A(t)=11⋅(0.98)tA(t)=11 \cdot(0.98)^{t}A(t)=11⋅(0.98)t. By what factor does the area decrease in 10 years?

Problem 39

Find the solutions to x−xe5x+2=0x-x e^{5 x+2}=0x−xe5x+2=0. Then find the second derivative of f(x)f(x)f(x) at x=1x=1x=1, given f′(x)+5f(x2)=f2(x)f'(x)+5f(x^2)=f^2(x)f′(x)+5f(x2)=f2(x) and f(1)=2f(1)=2f(1)=2.

Problem 40

Find the largest interval III over which the solution y(x)=1x2+Cy(x) = \frac{1}{x^2 + C}y(x)=x2+C1​ to the ODE y′+2x6=0y' + 2x^6 = 0y′+2x6=0 is defined, given y(4)=17y(4) = \frac{1}{7}y(4)=71​.

Problem 41

Find the carrying capacity of a population that grows according to the logistic equation dPdt=0.084P−0.00168P2\frac{dP}{dt} = 0.084P - 0.00168P^2dtdP​=0.084P−0.00168P2, where ttt is measured in weeks.

Problem 42

Find the equation for water level $L$ receding at 0.5 ft/day when initial level is 34 ft. Determine how many days until level reaches 26 ft.

Find the time it takes for a tranquilizer with a half-life of 50 hours to decay to $95\%$ of its original dosage using the exponential decay model $A=A_{0} e^{k t}$ .

Find the value of $y_0(2)$ where $y_0$ is the solution to the ODE $xy' + 2y = 2x$ with $y(1) = 3$ .

Find the amount of radioactive substance present (in grams) after 2 years, given the formula $y=8,000(2)^{-0.3 t}$ , where $t$ is in months. Round the result to three decimal places if needed.

Classify the equilibrium at the origin in $X = CX$ where $C = \begin{pmatrix} 9 & -11 \\ -11 & 9 \end{pmatrix}$ as a sink, saddle, or source.

Solve the ordinary differential equation $y-3x + (4y+3x)\frac{dy}{dx}=0$ using the homogeneous method.

Find all solutions to the second-order differential equation $y^{\prime \prime}=x \cos(x)$ .

Find the rat population in 2000 given the formula $n(t)=86 e^{0.04 t}$ where $t$ is years since 2000 and $n(t)$ is in millions.

Insect population of $\mathrm{P}(t) = 600e^{0.02t}$ at time $t$ days. Find: (a) population at $t=0$ , (b) growth rate, (c) graph, (d) population after 10 days, (e) when population reaches 900, (f) when population doubles.

Solve the differential equation $\frac{dy}{dx} = \frac{1}{5} \sqrt{y} \cos^2 \sqrt{y}$ .

Solve for t in the equation $h = 7 + 35t - 16t^2$ , given that $h = 25$ .

Find the solution to the ODE $y' = -1.7xy$ with $y(0) = 9.6$ , and evaluate the solution at $x = 1.3$ .

Find $\Delta T$ in the equation $q = m \times \Delta T \times C$ , where $q$ , $m$ , and $C$ are known.

Solve for the general solution of the differential equation $(x^2-xy+y^2)dx-xy\,dx=0$ . If homogeneous, use $y=vx$ .

Find the integrating factor for the first-order linear ODE $y' = -y + \cos x$ .

Find the general solution for the differential equation $\frac{d A}{d t}=-7 A$ , where $A(t)=Ce^{-7t}$ .

Solve the ODE $(6+t) \frac{d u}{d t} + u = 6+t$ for $t > 0$ .

Find the limit of the solution to the differential equation $y' = y^3 - 4y$ with initial condition $y(0) = 0.2$ as $t$ approaches infinity.

Use Euler's method with step size 0.1 to find $y(0.2)$ , where $y(x)$ satisfies $y'=2x+3y^2, y(0)=0$ .

Find $\frac{dx}{dt}$ when $x=-2$ and $y=2$ given $y^2 + xy - 3x = 6$ and $\frac{dy}{dt} = -2$ .

(a) Find depth where light intensity is half of surface intensity $L$ . Depth = $D$ meters. (b) Light intensity at 11 m depth = $I = L \cdot e^{(-1.95 \cdot 11)}$ . (c) Find depth where light intensity is 2.4% of surface intensity $L$ . Depth = $D$ meters.

Find the general solution of the second-order linear ODE $y'' + y = 0$ .

Find the solution of the initial value problem $g'(x)=8x(x^7-1/8); g(1)=1$ . The solution is $g(x)=\square$ .

Solve the ordinary differential equation $y'-2y=0$ .

Cooling liquid temperature model: $T(t) = 80(\frac{1}{2})^{\frac{t}{10}} + 20$ . Sketch graph, find temp. after 40 mins, time to cool to 45 degrees.

Find the initial number of people ill with the flu, given the function $N(t)=\frac{5000}{1+999 e^{-t}}$ models the number of people ill $t$ weeks after the outbreak. Round the result to the nearest person.

Find an exponential model $P=a\cdot b^t$ for a population of 16,000 organisms growing by $6.9\%$ annually.

Solve the initial value problem $\frac{dv}{dt}=\frac{2}{t\sqrt{t^2-1}}, t>1, v(2)=0$ and find $v(t)$ .

Find the height of an aircraft with atmospheric pressure $p=259$ mm Hg using the formula $p=760 e^{-0.145 \mathrm{h}}$ . Round the height to two decimal places.

Find the value of a quantity that decays exponentially at $8.5\%$ every 9 weeks, starting at 390, after 42 days, to the nearest hundredth.

Sketch direction field and solutions for differential equation $\frac{dy}{dx} = y^2 - x$ using isoclines for slopes $m = -1, 0, 1, 2$ .

Find the general solution for the third-order linear ODE $y'''- y''- y' + y = g(t)$ .

The two airplanes will be at the same altitude in $\sqrt{\frac{46400}{3200 + 2600}}$ minutes, when their altitude will be $23200$ feet.

A lake covers 11 square km, decreasing exponentially by 2% per year: $A(t)=11 \cdot(0.98)^{t}$ . By what factor does the area decrease in 10 years?

Find the solutions to $x-x e^{5 x+2}=0$ . Then find the second derivative of $f(x)$ at $x=1$ , given $f'(x)+5f(x^2)=f^2(x)$ and $f(1)=2$ .

Find the largest interval $I$ over which the solution $y(x) = \frac{1}{x^2 + C}$ to the ODE $y' + 2x^6 = 0$ is defined, given $y(4) = \frac{1}{7}$ .

Find the carrying capacity of a population that grows according to the logistic equation $\frac{dP}{dt} = 0.084P - 0.00168P^2$ , where $t$ is measured in weeks.

Exposure time to reach coma-inducing CO level: $T=.0002 x^{2}-.316 x+127.9$ , where $x$ is CO in ppm. Find exposure time when $x=560$ ppm. Round to nearest tenth.

Solve the separable differential equation $\frac{dy}{dx} + 4xy^2 = 0$ .

Solve the differential equation $y' = \frac{\sin \sqrt{x}}{7 \sqrt{x}}$ .

Solve the differential equation $\frac{d x}{d t}=5 t^{-1}+5$ for the given initial conditions.

Solve the initial value problem: $\frac{d y}{d x}=8 x e^{-6 y}, \quad y(0)=0$ .

Identify the exact ordinary differential equation (ODE) from the given options.
$(x+y) dx + (y+x) dy = 0$ $(xy+y) dx + (xy+x) dy = 0$ $(y+2) dx + (2x+1) dy = 0$

Solve the differential equation $3 y y' = 5 x$ .

Find $T_2$ given $t_2 \cdot \frac{P_1 V_1}{T_1} = \frac{P_2 V_2}{T_2}$ .

Find the solution to the differential equation $\frac{d y}{d x}=e^{2 x}+2 y$ . Options: A. $y=e^{2 x}$ , B. $y=-e^{2 x}$ , C. $y=x e^{2 x}$ , D. $y=-x e^{2 x}$ .

Radioactive fallout from a 1981 nuclear lab explosion is described by $f(x) = 1000(0.5)^{\frac{x}{30}}$ , where $f(x)$ is the remaining amount (kg) $x$ years later. Determine $f(40)$ and if the area will be safe by 2021.

Find the Laplace transform of $3te^{5t}$ using a table of Laplace transforms. Integration by parts may be useful.

Solve the differential equation: $x y \, dx - (x+2) \, dy = 0$ .

Un point $\mathrm{P}$ de masse $\mathrm{m}$ se déplace en coordonnées polaires. Trouvez les équations du mouvement et les expressions pour $\mathrm{r}(t)$ et $\omega$ quand $\omega$ est constant.

Find the function $f(x)$ given that $f^{\prime}(x)=2 f(x)$ and $f(2)=1$ .

Analyze the stability of equilibrium points for $\ddot{x}=(x-a)(x^{2}-a)$ across all real values of $a$ .

Solve the differential equation: $y^{\prime \prime}-7 y^{\prime}+12 y=0$ .

Find the salt concentration in a 100-gallon tank after 30 mins, starting with 10 lbs salt and adding brine at $4 \mathrm{gal/min}$ .

Upton Chuck free falls for 2.60 seconds. Calculate his final velocity and the distance fallen. Use $v = gt$ and $d = \frac{1}{2}gt^2$ .

Solve the DE: $(1-x) y' = y^2$ and determine if the solution is general or particular. Also solve: $a^2 dx = y \sqrt{y^2 - a^2} dy$ .

Solve the differential equation $y' = x e^{-y - x^2}$ . Is the solution general or particular?

Solve the separable differential equation $x y^{3} dx + (y+1)e^{-x} dy = 0$ and identify the type of solution. Also, solve $y' = x y^{3}$ .

Solve the differential equation: $\frac{dy}{dx} = 3y$ .

Solve the differential equation $\frac{d y}{d x}=\frac{3 y-7 x+7}{3 x-7 y-3}$ .

Solve the differential equation: $y' - \frac{2x-1}{x^2-x} y = 1$ .

Solve the differential equation: $y' - \frac{x^2 x - 1}{x^2 - x} y = 1$ .

A ball is dropped from a 94 ft building. Find the time to fall half the distance and to ground level: (a) $t=$ sec, (b) $t=$ sec.

A rumor spreads in a school. Given $y^{\prime}(t)=k y(1-y)$ , find when $90\%$ of 1000 students hear it.

Solve the logistic equation $\frac{d y}{d x}=5 y\left(1-\frac{y}{10}\right)$ for $y$ and find $y(0)=6$ .

Find the function $y(x)=7+\int_{2}^{x} \sin(t^{2}) dt$ given $\frac{d y}{d x}=\sin(x^{2})$ and $y(2)=7$ .