Chapter 7

For each of the following differential equations: a. Solve the initial value problem. b. [T] Use a graphing utility to graph the particular solution. $$ y^{\prime \prime}+5 y^{\prime}+15 y=0 \quad y(0)=-2, \quad y^{\prime}(0)=7 $$

(Principle of superposition) Prove that if \(y_{1}(x)\) and \(y_{2}(x)\) are solutions to a linear homogeneous differential equation, \(y^{\prime \prime}+p(x) y^{\prime}+q(x) y=0,\) then the function \(y(x)=c_{1} y_{1}(x)+c_{2} y_{2}(x), \quad\) where \(\quad c_{1}\) and \(c_{2}\) are constants, is also a solution.

Prove that if \(a, b,\) and \(c\) are positive constants, then all solutions to the second-order linear differential equation \(a y^{\prime \prime}+b y^{\prime}+c y=0\) approach zero as \(x \rightarrow \infty .\) (Hint: Consider three cases: two distinct roots, repeated real roots, and complex conjugate roots.)

Solve the following equations using the method of undetermined coefficients. $$ 2 y^{\prime \prime}-5 y^{\prime}-12 y=6 $$

Solve the following equations using the method of undetermined coefficients. $$ 3 y^{\prime \prime}+y^{\prime}-4 y=8 $$

Solve the following equations using the method of undetermined coefficients. $$ y^{\prime \prime}-6 y^{\prime}+5 y=e^{-x} $$

Solve the following equations using the method of undetermined coefficients. $$ y^{\prime \prime}-4 y^{\prime}+4 y=8 x^{2}+4 x $$

Solve the following equations using the method of undetermined coefficients. $$ y^{\prime \prime}-2 y^{\prime}-3 y=\sin 2 x $$

Solve the following equations using the method of undetermined coefficients. $$ y^{\prime \prime}+2 y^{\prime}+y=\sin x+\cos x $$

Solve the following equations using the method of undetermined coefficients. $$ y^{\prime \prime}+9 y=e^{x} \cos x $$

Solve the following equations using the method of undetermined coefficients. $$ y^{\prime \prime}+y=3 \sin 2 x+x \cos 2 x $$

Solve the following equations using the method of undetermined coefficients. $$ y^{\prime \prime}+3 y^{\prime}-28 y=10 e^{4 x} $$

Solve the following equations using the method of undetermined coefficients. $$ y^{\prime \prime}+10 y^{\prime}+25 y=x e^{-5 x}+4 $$

a. Write the form for the particular solution \(y_{p}(x)\) for the method of undetermined coefficients. b. Use a computer algebra system to find a particular solution to the given equation. $$ y^{\prime \prime}-y^{\prime}-y=x+e^{-x} $$

a. Write the form for the particular solution \(y_{p}(x)\) for the method of undetermined coefficients. b. Use a computer algebra system to find a particular solution to the given equation. $$ y^{\prime \prime}-3 y=x^{2}-4 x+11 $$

a. Write the form for the particular solution \(y_{p}(x)\) for the method of undetermined coefficients. b. Use a computer algebra system to find a particular solution to the given equation. $$ y^{\prime \prime}-y^{\prime}-4 y=e^{x} \cos 3 x $$

a. Write the form for the particular solution \(y_{p}(x)\) for the method of undetermined coefficients. b. Use a computer algebra system to find a particular solution to the given equation. $$ 2 y^{\prime \prime}-y^{\prime}+y=\left(x^{2}-5 x\right) e^{-x} $$

a. Write the form for the particular solution \(y_{p}(x)\) for the method of undetermined coefficients. b. Use a computer algebra system to find a particular solution to the given equation. $$ 4 y^{\prime \prime}+5 y^{\prime}-2 y=e^{2 x}+x \sin x $$

a. Write the form for the particular solution \(y_{p}(x)\) for the method of undetermined coefficients. b. Use a computer algebra system to find a particular solution to the given equation. $$ y^{\prime \prime}-y^{\prime}-2 y=x^{2} e^{x} \sin x $$

a. Write the form for the particular solution \(y_{p}(x)\) for the method of undetermined coefficients. b. Use a computer algebra system to find a particular solution to the given equation. $$ y^{\prime \prime}+3 y^{\prime}-4 y=2 e^{x} $$

a. Write the form for the particular solution \(y_{p}(x)\) for the method of undetermined coefficients. b. Use a computer algebra system to find a particular solution to the given equation. $$ y^{\prime \prime}+2 y^{\prime}=e^{3 x} $$

a. Write the form for the particular solution \(y_{p}(x)\) for the method of undetermined coefficients. b. Use a computer algebra system to find a particular solution to the given equation. $$ y^{\prime \prime}+6 y^{\prime}+9 y=e^{-x} $$

a. Write the form for the particular solution \(y_{p}(x)\) for the method of undetermined coefficients. b. Use a computer algebra system to find a particular solution to the given equation. $$ y^{\prime \prime}+2 y^{\prime}-8 y=6 e^{2 x} $$

Solve the differential equation using the method of variation of parameters. $$ 4 y^{\prime \prime}+y=2 \sin x $$

Solve the differential equation using the method of variation of parameters. $$ y^{\prime \prime}-9 y=8 x $$

Solve the differential equation using the method of variation of parameters. $$ y^{\prime \prime}+y=\sec x, \quad 0

Solve the differential equation using the method of variation of parameters. $$ y^{\prime \prime}+4 y=3 \csc 2 x, \quad 0

Find the unique solution satisfying the differential equation and the initial conditions given, where \(y_{p}(x)\) is the particular solution. \(y^{\prime \prime}-2 y^{\prime}+y=12 e^{x}, \quad y_{p}(x)=6 x^{2} e^{x}\) \(y(0)=6, \quad y^{\prime}(0)=0\)

Find the unique solution satisfying the differential equation and the initial conditions given, where \(y_{p}(x)\) is the particular solution. \( y^{\prime \prime}-7 y^{\prime}=4 x e^{7 x}, \quad y_{p}(x)=\frac{2}{7} x^{2} e^{7 x}-\frac{4}{49} x e^{7 x}\) \(y(0)=-1, \quad y^{\prime}(0)=0\)

Find the unique solution satisfying the differential equation and the initial conditions given, where \(y_{p}(x)\) is the particular solution. \(y^{\prime \prime}+y=\cos x-4 \sin x\), \(y_{p}(x)=2 x \cos x+\frac{1}{2} x \sin x, \quad y(0)=8, \quad y^{\prime}(0)=-4\)

Find the unique solution satisfying the differential equation and the initial conditions given, where \(y_{p}(x)\) is the particular solution. \(y^{\prime \prime}-5 y^{\prime}=e^{5 x}+8 e^{-5 x}\), \(y_{p}(x)=\frac{1}{5} x e^{5 x}+\frac{4}{25} e^{-5 x}, \quad y(0)=-2, \quad y^{\prime}(0)=0\)

Two linearly independent solutions \(-y_{1}\) and \(y_{2}\) -are given that satisfy the corresponding homogeneous equation. Use the method of variation of parameters to find a particular solution to the given nonhomogeneous equation. Assume \(x>0\) in each exercise. \(x^{2} y^{\prime \prime}+2 x y^{\prime}-2 y=3 x\), \(y_{1}(x)=x, \quad y_{2}(x)=x^{-2}\)

Two linearly independent solutions \(-y_{1}\) and \(y_{2}\) -are given that satisfy the corresponding homogeneous equation. Use the method of variation of parameters to find a particular solution to the given nonhomogeneous equation. Assume \(x>0\) in each exercise. \(x^{2} y^{\prime \prime}-2 y=10 x^{2}-1\), \(y_{1}(x)=x^{2}, \quad y_{2}(x)=x^{-1}\)

A 400-g mass stretches a spring \(5 \mathrm{~cm}\). Find the equation of motion of the mass if it is released from rest from a position \(15 \mathrm{~cm}\) below the equilibrium position. What is the frequency of this motion?

A block has a mass of \(9 \mathrm{~kg}\) and is attached to a vertical spring with a spring constant of \(0.25 \mathrm{~N} / \mathrm{m}\). The block is stretched \(0.75 \mathrm{~m}\) below its equilibrium position and released. a. Find the position function \(x(t)\) of the block. b. Find the period and frequency of the vibration. c. Sketch a graph of \(x(t)\). d. At what time does the block first pass through the equilibrium position?

A block has a mass of \(5 \mathrm{~kg}\) and is attached to a vertical spring with a spring constant of \(20 \mathrm{~N} / \mathrm{m}\). The block is released from the equilibrium position with a downward velocity of \(10 \mathrm{~m} / \mathrm{sec}\) a. Find the position function \(x(t)\) of the block. b. Find the period and frequency of the vibration. c. Sketch a graph of \(x(t)\). d. At what time does the block first pass through the equilibrium position?

A 1 -kg mass is attached to a vertical spring with a spring constant of \(21 \mathrm{~N} / \mathrm{m}\). The resistance in the springmass system is equal to 10 times the instantaneous velocity of the mass. a. Find the equation of motion if the mass is released from a position \(2 \mathrm{~m}\) below its equilibrium position with a downward velocity of \(2 \mathrm{~m} / \mathrm{sec}\). b. Graph the solution and determine whether the motion is overdamped, critically damped, or underdamped.

An 800 -lb weight ( 25 slugs) is attached to a vertical spring with a spring constant of \(226 \mathrm{lb} / \mathrm{ft}\). The system is immersed in a medium that imparts a damping force equal to 10 times the instantaneous velocity of the mass. a. Find the equation of motion if it is released from a position \(20 \mathrm{ft}\) below its equilibrium position with a downward velocity of \(41 \mathrm{ft} / \mathrm{sec}\). b. Graph the solution and determine whether the motion is overdamped, critically damped, or underdamped.

A 9-kg mass is attached to a vertical spring with a spring constant of \(16 \mathrm{~N} / \mathrm{m}\). The system is immersed in a medium that imparts a damping force equal to 24 times the instantaneous velocity of the mass. a. Find the equation of motion if it is released from its equilibrium position with an upward velocity of 4 \(\mathrm{m} / \mathrm{sec}\). b. Graph the solution and determine whether the motion is overdamped, critically damped, or underdamped.

A 1 -kg mass stretches a spring \(6.25 \mathrm{~cm}\). The resistance in the spring-mass system is equal to eight times the instantaneous velocity of the mass. a. Find the equation of motion if the mass is released from a position \(5 \mathrm{~m}\) below its equilibrium position with an upward velocity of \(10 \mathrm{~m} / \mathrm{sec}\). b. Determine whether the motion is overdamped, critically damped, or underdamped.

A 32-lb weight (1 slug) stretches a vertical spring 4 in. The resistance in the spring-mass system is equal to four times the instantaneous velocity of the mass. a. Find the equation of motion if it is released from its equilibrium position with a downward velocity of \(12 \mathrm{ft} / \mathrm{sec}\). b. Determine whether the motion is overdamped, critically damped, or underdamped.

A mass that weighs \(8 \mathrm{lb}\) stretches a spring 6 inches. The system is acted on by an external force of \(8 \sin 8 t\) lb. If the mass is pulled down 3 inches and then released, determine the position of the mass at any time.

A mass that weighs 6 lb stretches a spring 3 in. The system is acted on by an external force of \(8 \sin (4 t) \mathrm{lb}\). If the mass is pulled down 1 inch and then released, determine the position of the mass at any time.

Find the charge on the capacitor in an \(R L C\) series circuit where \(L=40 \mathrm{H}, R=30 \Omega, \quad C=1 / 200 \mathrm{~F},\) and \(E(t)=200 \mathrm{~V}\). Assume the initial charge on the capacitor is \(7 \mathrm{C}\) and the initial current is \(0 \mathrm{~A}\).

Find the charge on the capacitor in an \(R L C\) series circuit where \(L=2 \mathrm{H}, R=24 \Omega, \quad C=0.005 \mathrm{~F},\) and \(E(t)=12 \sin 10 t \quad \mathrm{~V} .\) Assume the initial charge on the capacitor is \(0.001 \mathrm{C}\) and the initial current is \(0 \mathrm{~A}\).

A series circuit consists of a device where \(L=1\) \(\mathrm{H}, \quad R=20 \Omega, \quad C=0.002 \mathrm{~F},\) and \(E(t)=12 \mathrm{~V} .\) If the initial charge and current are both zero, find the charge and current at time \(t\).

Find a power series solution for the following differential equations. $$ y^{\prime \prime}+6 y^{\prime}=0 $$

Find a power series solution for the following differential equations. $$ 5 y^{\prime \prime}+y^{\prime}=0 $$

Find a power series solution for the following differential equations. $$ y^{\prime \prime}-y=0 $$

Find a power series solution for the following differential equations. $$ y^{\prime}-2 x y=0 $$