University Calculus: Early Transcendentals 4th

Joel Hass, Christopher Heil, Przemyslaw Bogacki

Chapter 17

Second-Order Differential Equations

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Problem 1

Find the general solution of the given equation.
$$y^{\prime \prime}-y^{\prime}-12 y=0$$

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Problem 2

Find the general solution of the given equation.
$$3 y^{\prime \prime}-y^{\prime}=0$$

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Problem 3

Find the general solution of the given equation.
$$y^{\prime \prime}+3 y^{\prime}-4 y=0$$

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Problem 4

Find the general solution of the given equation.
$$y^{\prime \prime}-9 y=0$$

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Problem 5

Find the general solution of the given equation.
$$y^{\prime \prime}-4 y=0$$

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Problem 6

Find the general solution of the given equation.
$$y^{\prime \prime}-64 y=0$$

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Problem 7

Find the general solution of the given equation.
$$2 y^{\prime \prime}-y^{\prime}-3 y=0$$

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Problem 8

Find the general solution of the given equation.
$$9 y^{\prime \prime}-y=0$$

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Problem 9

Find the general solution of the given equation.
$$8 y^{\prime \prime}-10 y^{\prime}-3 y=0$$

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Problem 10

Find the general solution of the given equation.
$$3 y^{\prime \prime}-20 y^{\prime}+12 y=0$$

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Problem 11

Find the general solution of the given equation.
$$v^{\prime \prime}+9 v=0$$

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Problem 12

Find the general solution of the given equation.
$$y^{\prime \prime}+4 y^{\prime}+5 y=0$$

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Problem 13

Find the general solution of the given equation.
$$y^{\prime \prime}+25 y=0$$

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Problem 14

Find the general solution of the given equation.
$$y^{\prime \prime}+y=0$$

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Problem 15

Find the general solution of the given equation.
$$y^{\prime \prime}-2 y^{\prime}+5 y=0$$

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Problem 16

Find the general solution of the given equation.
$$y^{\prime \prime}+16 y=0$$

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Problem 17

Find the general solution of the given equation.
$$y^{\prime \prime}+2 y^{\prime}+4 y=0$$

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Problem 18

Find the general solution of the given equation.
$$y^{\prime \prime}-2 y^{\prime}+3 y=0$$

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Problem 19

Find the general solution of the given equation.
$$y^{\prime \prime}+4 y^{\prime}+9 y=0$$

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Problem 20

Find the general solution of the given equation.
$$4 y^{\prime \prime}-4 y^{\prime}+13 y=0$$

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Problem 21

Find the general solution of the given equation.
$$y^{\prime \prime}=0$$

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Problem 22

Find the general solution of the give equation.
$$y^{\prime \prime}+8 y^{\prime}+16 y=0$$

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Problem 23

Find the general solution of the given equation.
$$\frac{d^{2} y}{d x^{2}}+4 \frac{d y}{d x}+4 y=0$$

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Problem 24

Find the general solution of the given equation.
$$\frac{d^{2} y}{d x^{2}}-6 \frac{d y}{d x}+9 y=0$$

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Problem 25

Find the general solution of the given equation.
$$\frac{d^{2} y}{d x^{2}}+6 \frac{d y}{d x}+9 y=0$$

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Problem 26

Find the general solution of the given equation.
$$4 \frac{d^{2} y}{d x^{2}}-12 \frac{d y}{d x}+9 y=0$$

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Problem 27

Find the general solution of the given equation.
$$4 \frac{d^{2} y}{d x^{2}}+4 \frac{d y}{d x}+y=0$$

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Problem 28

Find the general solution of the given equation.
$$4 \frac{d^{2} y}{d x^{2}}-4 \frac{d y}{d x}+y=0$$

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Problem 29

Find the general solution of the given equation.
$$9 \frac{d^{2} y}{d x^{2}}+6 \frac{d y}{d x}+y=0$$

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Problem 30

Find the general solution of the given equation.
$$9 \frac{d^{2} y}{d x^{2}}-12 \frac{d y}{d x}+4 y=0$$

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Problem 31

Find the unique solution of the second-order initial value problem.
$$y^{\prime \prime}+6 y^{\prime}+5 y=0, \quad y(0)=0, y^{\prime}(0)=3$$

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Problem 32

Find the unique solution of the second-order initial value problem.
$$y^{\prime \prime}+16 y=0, \quad y(0)=2, y^{\prime}(0)=-2$$

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Problem 33

Find the unique solution of the second-order initial value problem.
$$y^{\prime \prime}+12 y=0, \quad y(0)=0, y^{\prime}(0)=1$$

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Problem 34

Find the unique solution of the second-order initial value problem.
$$12 y^{\prime \prime}+5 y^{\prime}-2 y=0, \quad y(0)=1, y^{\prime}(0)=-1$$

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Problem 35

Find the unique solution of the second-order initial value problem.
$$y^{\prime \prime}+8 y=0, \quad y(0)=-1, y^{\prime}(0)=2$$

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Problem 36

Find the unique solution of the second-order initial value problem.
$$y^{\prime \prime}+4 y^{\prime}+4 y=0, \quad y(0)=0, y^{\prime}(0)=1$$

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Problem 37

Find the unique solution of the second-order initial value problem.
$$y^{\prime \prime}-4 y^{\prime}+4 y=0, \quad y(0)=1, y^{\prime}(0)=0$$

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Problem 38

Find the unique solution of the second-order initial value problem.
$$4 y^{\prime \prime}-4 y^{\prime}+y=0, \quad y(0)=4, y^{\prime}(0)=4$$

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Problem 39

Find the unique solution of the second-order initial value problem.
$$4 \frac{d^{2} y}{d x^{2}}+12 \frac{d y}{d x}+9 y=0, \quad y(0)=2, \frac{d y}{d x}(0)=1$$

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Problem 40

Find the unique solution of the second-order initial value problem.
$$9 \frac{d^{2} y}{d x^{2}}-12 \frac{d y}{d x}+4 y=0, \quad y(0)=-1, \frac{d y}{d x}(0)=1$$

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Problem 41

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

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Problem 42

Find the general solution.
$$6 y^{\prime \prime}-y^{\prime}-y=0$$

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Problem 43

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

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Problem 44

Find the general solution.
$$9 y^{\prime \prime}+12 y^{\prime}+4 y=0$$

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Problem 45

Find the general solution.
$$4 y^{\prime \prime}+20 y=0$$

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Problem 46

Find the general solution.
$$y^{\prime \prime}+2 y^{\prime}+2 y=0$$

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Problem 47

Find the general solution.
$$25 y^{\prime \prime}+10 y^{\prime}+y=0$$

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Problem 48

Find the general solution.
$$6 y^{\prime \prime}+13 y^{\prime}-5 y=0$$

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Problem 49

Find the general solution.
$$4 y^{\prime \prime}+4 y^{\prime}+5 y=0$$

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Problem 50

Find the general solution.
$$y^{\prime \prime}+4 y^{\prime}+6 y=0$$

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Problem 51

Find the general solution.
$$16 y^{\prime \prime}-24 y^{\prime}+9 y=0$$

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Problem 52

Find the general solution.
$$6 y^{\prime \prime}-5 y^{\prime}-6 y=0$$

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Problem 53

Find the general solution.
$$9 y^{\prime \prime}+24 y^{\prime}+16 y=0$$

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Problem 54

Find the general solution.
$$4 y^{\prime \prime}+16 y^{\prime}+52 y=0$$

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Problem 55

Find the general solution.
$$6 y^{\prime \prime}-5 y^{\prime}-4 y=0$$

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Problem 56

Solve the initial value problem.
$$y^{\prime \prime}-2 y^{\prime}+2 y=0, \quad y(0)=0, y^{\prime}(0)=2$$

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Problem 57

Solve the initial value problem.
$$y^{\prime \prime}+2 y^{\prime}+y=0, \quad y(0)=1, y^{\prime}(0)=1$$

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Problem 58

Solve the initial value problem.
$$4 y^{\prime \prime}-4 y^{\prime}+y=0, \quad y(0)=-1, y^{\prime}(0)=2$$

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Problem 59

Solve the initial value problem.
$$3 y^{\prime \prime}+y^{\prime}-14 y=0, \quad y(0)=2, y^{\prime}(0)=-1$$

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Problem 60

Solve the initial value problem.
$$4 y^{\prime \prime}+4 y^{\prime}+5 y=0, \quad y(\pi)=1, y^{\prime}(\pi)=0$$

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Problem 61

Prove that the two solution functions in Theorem 3 are linearly independent.

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Problem 62

Prove that the two solution functions in Theorem 4 are linearly independent.

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Problem 63

Prove that the two solution functions in Theorem 5 are linearly independent.

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Problem 64

Prove that if $y_{1}$ and $y_{2}$ are linearly independent solutions to the homogeneous equation (2), then the functions $y_{3}=y_{1}+y_{2}$ and $y_{4}=y_{1}-y_{2}$ are also linearly independent solutions.

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Problem 65

a. Show that there is no solution to the boundary value problem
$$y^{\prime \prime}+4 y=0, \quad y(0)=0, y(\pi)=1
$$b. Show that there are infinitely many solutions to the boundary value problem
$$y^{\prime \prime}+4 y=0, \quad y(0)=0, y(\pi)=0$$

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Problem 66

Show that if $a, b,$ and $c$ are positive constants, then all solutions of the homogeneous differential equation
$$a y^{\prime \prime}+b y^{\prime}+c y=0$$approach zero as $x \rightarrow \infty$

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