Deck 35: Differential Equations

Question 1

Give the order and the degree of the equation, and tell whether it is ordinary or partial: $\frac{\partial y}{\partial x}+7+\frac{\partial^{2} y}{\partial x^{2}}=0$

Answer 1

Second order, first degree, partial

Question 2

Solve the differential equation: ${x+y^{\prime}}=9 x^{2}+5 x$

Answer 2

$y=3 x^{3}+2 x^{2}+C$

Question 3

Give the order and the degree of the equation, and tell whether it is ordinary or partial:
$\frac{d^{2} y}{d x^{2}}+3 \frac{d y}{d x}+2 y=1+2 e^{-x}$

Answer 3

Second order, first degree, ordinary

Question 4

Solve the differential equation: $3 \frac{d y}{d x}=4 x-x^{3}$

Question 5

Solve the differential equation: $2 d y+x^{2} d x=0$

Question 6

Is the function $4 y=C x^{3}$ a solution to the differential equation $\frac{d y}{d x}=\frac{3 y}{x}$ ?

Question 7

Is the function $y=2 C x^{2}$ a solution to the differential equation $y^{\prime \prime}+2 y^{\prime}+2 y=\frac{2 y(1+x)^{2}}{x^{2}}$ ?

Question 8

Use Euler's method to find an approximate solution to the differential equation $\frac{d y}{d x}=\frac{x^{2}}{y}$ from the boundary value $(1,1)$ to $x=5$ . Increase $x$ in steps of $\Delta x=1$ unit.

Question 9

Use Euler's method to graphically solve the differential equation $\frac{d y}{d x}=\frac{x^{2}}{y}$ from the boundary value $(1,1)$ to $x=5$ . Increase $x$ in steps of $\Delta x=1$ unit.

Question 10

Find the general solution: $y^{\prime}=\frac{3 x^{2}}{y}$

Question 11

Find the general solution: $y^{\prime}=\frac{3 y x^{2}}{x^{3}+1}$

Question 12

Find the general solution: $y d y=x^{2} d x$

Question 13

Find the general solution: $d y=\left(e^{2 x}+1\right) d x$

Question 14

Find the general solution: $\cot x d y-\sec y d x=0$

Question 15

Find the particular solution: $y^{\prime}\left(x^{2}-1\right)=2 x y(x=2$ when $y=9)$

Question 16

Find the general solution of the differential equation: $y^{\prime}=(1-y) \cos x$

Question 17

Find the particular solution of the differential equation: $y^{\prime}=(1-y) \cos x$ , when $y(\pi)=2$

Question 18

Find the general solution of the differential equation: $y^{\prime}-3 x^{2} y=0$

Question 19

Find the general solution: $\frac{d y}{d x}=\frac{5 x^{3}}{3 y^{2}}$

Question 20

Find the particular solution: $\sec x d y-\csc y d x=0, x=\pi$ when $y=\frac{\pi}{2}$

Question 21

Find the general solution: $e^{x} y y^{\prime}=1$

Question 22

Find the particular solution: $x y d y-x^{2} d x+1 d x=0, x=1$ when $y=2$

Question 23

Find the general solution: $\sqrt{x^{2}+1} d y=x y d x$

Question 24

Find the general solution: $\frac{x}{4}-y=x y^{\prime}$

Question 25

Find the general solution: $y\left(1-y^{2}\right)=y^{\prime}\left(x+x y^{2}\right)$

Question 26

Find the general solution: $\left(x-3 x^{3}\right) y^{\prime}-3 x^{2} y-y=0$

Question 27

Find the general solution to the first-order differential equation: $y-x y^{\prime}=y^{2} e^{x}$

Question 28

Find the particular solution of the differential equation: $\left(y^{2} \sin x-x-y^{3}\right) d x+\left(2 y \cos x-3 x y^{2}\right) d y=0$ at $y(0)=1$ .

Question 29

Find the general solution: $y d x=4\left(x^{2}+y^{2}\right) d x+x d y$

Question 30

Find the general solution: $4 y d x=4 x d y+8 y^{2} d y$

Question 31

Find the general solution: $2 x y^{2} d x+2 x^{2} y d y-3 d x=0$

Question 32

Find the general solution: $2 x d y=2 y d x+5 x^{2} d y$

Question 33

Find the particular solution: $\left(y^{2}-x y\right) d x+\left(3 x^{2}-x y\right) d y=0 ; x=1$ and $y=3$

Question 34

Find the general solution of the differential equation: $\left(y^{2}+y x\right) d x-x^{2} d y=0$

Question 35

Find the general solution: $x^{2} y d x=\left(x^{3}+y^{3}\right) d y$

Question 36

Find the general solution: $\left(x^{2}+y^{2}\right) d x=x y d y$

Question 37

Find the particular solution: $(y-x) d x-3 x d y=0 ; x=-1$ and $y=1$

Question 38

Find the general solution: $e^{-x} y^{\prime}-y e^{-x}=1$

Question 39

Find the general solution: $\frac{2 d y}{d x}+\frac{4 y}{x}=6 \sin x$

Question 40

Find the particular solution: $\frac{d y}{d x}+2 y=\cos x ; x=0$ when $y=1$

Question 41

Find the general solution of the differential equation: $\left(1+x^{2}\right) y^{\prime}+2 x y=0$

Question 42

Find the particular solution of the differential equation: $\left(1+x^{2}\right) y^{\prime}+2 x y=1$ , when $y(0)=4$

Question 43

Find the particular solution of the differential equation: $y^{\prime}+2 x y=x$ , when $y(0)=3$

Question 44

Find the particular solution of the differential equation: $x^{2} y^{\prime}-2 x y=3 y^{4}$ , when $y(1)=0.5$

Question 45

Find the general solution: $x y^{\prime}-2 x^{2}=4 y$

Question 46

Find the particular solution: $y^{\prime}+y=4 \sin x, x=0$ when $y=2$

Question 47

Find the general solution: $x y^{\prime}+y=x^{3}$

Question 48

Find the general solution: $(x+2) d y+4 y d x=(x+2)^{2} d x$

Question 49

Find the equation of a curve whose slope is equal to the product of the abscissa squared and the ordinate, and which passes through the point $(3,1)$ .

Question 50

In the figure below, a curve passes through $(3,1)$ . The length of the normal $P C$ through any point to the $x$ -axis is equal to twice the ordinate of that point, for $y>0$ . The equation for the distance PC for any curve is given by $P C=y \sqrt{1+\left(y^{\prime}\right)^{2}}$ . Find the equation of this curve.

Question 51

Find the equation of the curve that passes through $(1,0)$ whose slope is $y' = \frac{y+1}{x}$

Question 52

Find the orthogonal trajectories of the family of curves $x=k y^{2}$ , where $k$ is a constant.

Question 53

Find the equation of a curve whose slope is equal to the abscissa cubed divided by the ordinate, and which passes through the point $(2,1)$ .

Question 54

Find the equation of the curve that passes through $(0,5)$ whose slope is $y^{\prime}=x-y$ .

Question 55

In nuclear decay the rate of disappearance of a particular active compound is directly proportional to the amount of compound left. Find an equation for the amount of compound left, $y$ , in terms of the constant of proportionality, $k$ , and time.

Question 56

A $22 \mathrm{~kg}$ rock is dropped from a boat on the ocean. It experiences water friction that is proportional to its speed down through the water. It reaches a maximum speed of $4.9 \mathrm{~m} / \mathrm{sec}$ . Find an equation for the acceleration of the rock.

Question 57

A certain bacteria grows exponentially at a rate of $15 \% /$ hour. Starting with 400 bacteria, how many will there be after $24 \mathrm{~h}$ ?

Question 58

A certain bacteria grows exponentially at a rate of $8 \% / h o u r$ . Starting with 100 bacteria, how much bacteria will there be after 24 hours? Round your answer to three significant digits.

Question 59

A metal ingot is taken out of a furnace when it reaches $950^{\circ} \mathrm{C}$ above room temperature. If it cools at $4 \% / minutes$ , find its temperature (above room temperature) 1 hour later. Round your answer to three significant digits.

Question 60

The switch in the figure below is closed at $t=0$ . If $E=4 \mathrm{~V}, \mathrm{R}=5.0$ ohms, and $L=3.30$ henrys, find the current through the inductor at $t=1 \mathrm{~s}$ .

Question 61

For the circuit shown the resistance decreases with time according to the expression: $R=\frac{6.00}{t+1}$
Find the equation for the current $i$ .

Question 62

For the circuit shown, $E=25 \mathrm{~V}, L=4 \mathrm{H}$ , and $R=250 \Omega$ . If the switch is thrown from 2 to 1 (charging), find the voltage across the inductance at $5 \mathrm{~ms}$ .

Question 63

Solve: $y^{\prime \prime}=e^{3 x}$

Question 64

Solve for $\mathrm{y}: y^{\prime \prime}-4 x^{2}=0$

Question 65

Solve for y: $y^{\prime \prime}=32 \cos 4 x$ , where $y^{\prime}=2$ at the point $(0,1)$

Question 66

Solve for $\mathrm{y}: y^{\prime \prime}=10 e^{2 x}$

Question 67

Solve for $\mathrm{y}: y^{\prime \prime}-2 x=0$ , where $y^{\prime}=\frac{1}{2}$ at the point $(1,1)$

Question 68

Find the general solution: $y^{\prime \prime}+4 y^{\prime}+3 y=0$

Question 69

Find the general solution: $5 y^{\prime \prime}+6 y^{\prime}+1 y=0$

Question 70

Find the general solution: $y^{\prime \prime}+6 y^{\prime}+9 y=0$

Question 71

Find the general solution: $y^{\prime \prime}+8 y=4 y^{\prime}$

Question 72

Find the particular solution: $y^{\prime \prime}=2 y^{\prime}+3 y, y^{\prime}=3$ at $(0,1)$

Question 73

Find the particular solution: $y^{\prime \prime}+8 y^{\prime}+20 y=0, y^{\prime}=2$ at $(0,3)$

Question 74

Find the general solution: ${ }^{9 y^{\prime \prime \prime}-12 y^{\prime \prime}-11 y^{\prime}-2 y=0}$

Question 75

Solve the differential equation: $y^{\prime \prime}+6 y^{\prime}+8 y=0$

Question 76

Solve the differential equation: $2 y^{\prime \prime}-5 y^{\prime}-3 y=0$ , for $y^{\prime}(0,1)=2$

Question 77

Solve the differential equation: $y^{\prime \prime}-6 y^{\prime}+9 y=0$ , when $y^{\prime}(0,1)=-2$

Question 78

Find the general solution: $2 y^{\prime \prime}+3 y^{\prime}-5 y=0$

Question 79

Find the particular solution: $4 y^{\prime \prime}-12 y^{\prime}+9 y=0, y=2$ and $y^{\prime}=8$ when $x=0$

Question 80

Solve the differential equation: $5 y^{\prime \prime \prime}-28 y^{\prime \prime}-12 y^{\prime}=0$