Chapter 7: Second-Order Differential EquationsContent check — Calculus Vol. 3, Chapter 7: Second-Order Differential Equations

Classify each of the following equations as linear or nonlinear. If the equation is linear, determine further whether it is homogeneous or nonhomogeneous.

a. ${(y^{″})}^2-$ $y^{\prime }+$ $8x^{3}y=$ $0$

b. $(\text{sin}t)y^{″}+$ $\text{cos}t-$ $3t{y}^{\prime }=$ $0$

Determine whether the following pairs of functions are linearly dependent or linearly independent: $f_1(x)=e^x,$ $f_2(x)=3e^{3x}.$

If $y_1(x)=e^{3x}$ and $y_2(x)=x{e}^{3x}$ are solutions to $y^{″}-$ $6y^{\prime }+$ $9y=$ $0,$ what is the general solution?

Find the general solution to the following differential equations:

a. $y^{″}-$ $2y^{\prime }+$ $10y=$ $0$

b. $y^{″}+$ $14y^{\prime }+$ $49y=$ $0$

Solve the initial-value problem $y^{″}-$ $3y^{\prime }-$ $10y=$ $0,$ $y(0)=0,$ $y^{\prime }(0)=7.$

Solve the following initial-value problem and graph the solution: $y^{″}-$ $2y^{\prime }+$ $10y=$ $0,$ $y(0)=$ $2,$ $y^{\prime }(0)=$ $\mathrm{-1}$

Suppose the following initial-value problem models the position (in feet) of a mass in a spring-mass system at any given time. Solve the initial-value problem and graph the solution. What is the position of the mass at time $t=0.3$ sec? How fast is it moving at time $t=0.1$ sec? In what direction?

$$y^{″}+14y^{\prime }+49y=0,y(0)=0,y^{\prime }(0)=1$$

Classify each of the following equations as linear or nonlinear. If the equation is linear, determine whether it is homogeneous or nonhomogeneous.

$x^3{y}^{″}+$ $(x-1)y^{\prime }-$ $8y=$ $0$

Classify each of the following equations as linear or nonlinear. If the equation is linear, determine whether it is homogeneous or nonhomogeneous.

$x{y}^{″}+e^y{y}^{\prime }=x$

Classify each of the following equations as linear or nonlinear. If the equation is linear, determine whether it is homogeneous or nonhomogeneous.

$y^{″}+$ $\left(\text{sin}x\right)y^{\prime }-$ $xy=$ $4y$

For each of the following problems, verify that the given function is a solution to the differential equation. Use a graphing utility to graph the particular solutions for several values of c₁ and c₂. What do the solutions have in common?

$y^{″}-$ $3y^{\prime }-$ $10y=$ $0$

For each of the following problems, verify that the given function is a solution to the differential equation. Use a graphing utility to graph the particular solutions for several values of c₁ and c₂. What do the solutions have in common?

$y^{″}+4y^{\prime }+4y=0$

For each of the following problems, verify that the given function is a solution to the differential equation. Use a graphing utility to graph the particular solutions for several values of c₁ and c₂. What do the solutions have in common?

$2y^{″}-$ $3y^{\prime }-$ $5y=$ $0$

For each of the following problems, verify that the given function is a solution to the differential equation. Use a graphing utility to graph the particular solutions for several values of c₁ and c₂. What do the solutions have in common?

$y^{″}+y^{\prime }+y=0$

For each of the following problems, verify that the given function is a solution to the differential equation. Use a graphing utility to graph the particular solutions for several values of c₁ and c₂. What do the solutions have in common?

$y^{″}-121y=0$

For each of the following problems, verify that the given function is a solution to the differential equation. Use a graphing utility to graph the particular solutions for several values of c₁ and c₂. What do the solutions have in common?

$y^{″}+81y=0$

For each of the following problems, verify that the given function is a solution to the differential equation. Use a graphing utility to graph the particular solutions for several values of c₁ and c₂. What do the solutions have in common?

$2y^{″}=0$

For each of the following problems, verify that the given function is a solution to the differential equation. Use a graphing utility to graph the particular solutions for several values of c₁ and c₂. What do the solutions have in common?

$3y^{″}-$ $2y^{\prime }-$ $7y=$ $0$

For each of the following problems, verify that the given function is a solution to the differential equation. Use a graphing utility to graph the particular solutions for several values of c₁ and c₂. What do the solutions have in common?

$36\frac{d^{2}y}{d{x}^2}+$ $12\frac{dy}{dx}+$ $y=$ $0$

For each of the following problems, verify that the given function is a solution to the differential equation. Use a graphing utility to graph the particular solutions for several values of c₁ and c₂. What do the solutions have in common?

$\frac{d^{2}y}{d{x}^2}-$ $9\frac{dy}{dx}=$ $0$

Solve the initial-value problem.

$y^{″}+$ $5y^{\prime }+$ $6y=$ $0,$ $y(0)=$ $0,$ $y^{\prime }(0)=$ $\mathrm{-2}$

Solve the initial-value problem.

$y^{″}+$ $4y=$ $0,$ $y(0)=$ $3,$ $y^{\prime }(0)=$ $10$

Solve the initial-value problem.

$y^{″}-$ $y^{\prime }-$ $30y=$ $0,$ $y(0)=$ $1,$ $y^{\prime }(0)=$ $\mathrm{-16}$

Solve the initial-value problem.

$25y^{″}+$ $10y^{\prime }+$ $y=$ $0,$ $y(0)=$ $2,$ $y^{\prime }(0)=$ $1$

Solve the boundary-value problem, if possible.

$y^{″}+$ $y^{\prime }-$ $42y=$ $0,$ $y(0)=$ $0,$ $y(1)=$ $2$

Solve the boundary-value problem, if possible.

$y^{″}+$ $10y^{\prime }+$ $34y=$ $0,$ $y(0)=$ $6,$ $y(\pi )=$ $2$

Solve the boundary-value problem, if possible.

$y^{″}-$ $4y^{\prime }+$ $4y=$ $0,$ $y(0)=$ $2,$ $y(1)=$ $\mathrm{-1}$

Solve the boundary-value problem, if possible.

$y^{″}+$ $9y=$ $0,$ $y(0)=$ $4,$ $y\left(\frac{\pi }{3}\right)=$ $\mathrm{-4}$

Solve the boundary-value problem, if possible.

Find a differential equation with a general solution that is $y=$ $c_1{e}^{x\text{/}5}+$ $c_2{e}^{\mathrm{-4}x}.$

For each of the following differential equations:

$y^{″}+$ $64y=$ $0;$ $y(0)=$ $3,$ $y^{\prime }(0)=$ $\mathrm{–24}$

For each of the following differential equations:

$y^{″}+$ $5y^{\prime }+$ $15y=$ $0$ $y(0)=$ $\mathrm{-2},$ $y^{\prime }(0)=$ $7$

Given that $y_p(x)=\mathrm{-2}$ is a particular solution to $y^{″}-$ $3y^{\prime }-$ $4y=$ $8,$ write the general solution and verify that the general solution satisfies the equation.

Find the general solution to $y^{″}-$ $4y^{\prime }+$ $4y=$ $7\text{sin}t-$ $\text{cos}t\text{.}$

Find the general solution to the following differential equations.

a. $y^{″}-$ $5y^{\prime }+$ $4y=$ $3e^x$

b. $y^{″}+$ $y^{\prime }-$ $6y=$ $52\text{cos}2t$

Use Cramer’s rule to solve the following system of equations.

$$\begin{array}{ccc}\hfill 2x{z}_1-3z_2& =\hfill & 0\hfill \\ \hfill x^2{z}_1+4x{z}_2& =\hfill & x+1\hfill \end{array}$$

Find the general solution to the following differential equations.

a. $y^{″}+y=\text{sec}x$

b. $x^{″}-$ $2x^{\prime }+$ $x=$ $\frac{e^t}{t}$

Solve the following equations using the method of undetermined coefficients.

$3y^{″}+y^{\prime }-4y=8$

Solve the following equations using the method of undetermined coefficients.

$y^{″}+16y=e^{\mathrm{-2}x}$

Solve the following equations using the method of undetermined coefficients.

$y^{″}-$ $4y^{\prime }+$ $4y=$ $8x^2+$ $4x$

Solve the following equations using the method of undetermined coefficients.

$y^{″}+$ $2y^{\prime }+$ $y=$ $\text{sin}x+$ $\text{cos}x$

Solve the following equations using the method of undetermined coefficients.

$y^{″}+$ $y=$ $3\text{sin}2x+$ $x\text{cos}2x$

Solve the following equations using the method of undetermined coefficients.

$y^{″}+$ $10y^{\prime }+$ $25y=$ $x{e}^{\mathrm{-5}x}+$ $4$

In each of the following problems,

$y^{″}-$ $3y=$ $x^2-$ $4x+$ $11$

In each of the following problems,

$2y^{″}-$ $y^{\prime }+$ $y=$ $(x^2-5x)e^{\text{−}x}$

In each of the following problems,

$y^{″}-$ $y^{\prime }-$ $2y=$ $x^2{e}^x\text{sin}x$

Solve the differential equation using either the method of undetermined coefficients or the variation of parameters.

$y^{″}+2y^{\prime }=e^{3x}$

Solve the differential equation using either the method of undetermined coefficients or the variation of parameters.

$y^{″}+$ $2y^{\prime }-$ $8y=$ $6e^{2x}$

Solve the differential equation using the method of variation of parameters.

$y^{″}-9y=8x$

Solve the differential equation using the method of variation of parameters.

$y^{″}+$ $4y=$ $3\text{csc}2x,$ $0<$ $x<$ $\pi \text{/}2$

Find the unique solution satisfying the differential equation and the initial conditions given, where $y_p(x)$ is the particular solution.

$y^{″}-$ $7y^{\prime }=$ $4x{e}^{7x},$ $y_p(x)=$ $\frac{2}{7}{x}^2{e}^{7x}-$ $\frac{4}{49}x{e}^{7x},$ $y(0)=$ $\mathrm{-1},$ $y\prime (0)=$ $0$

Find the unique solution satisfying the differential equation and the initial conditions given, where $y_p(x)$ is the particular solution.

$y^{″}-$ $5y^{\prime }=$ $e^{5x}+$ $8e^{\mathrm{-5}x},$ $y_p(x)=$ $\frac{1}{5}x{e}^{5x}+$ $\frac{4}{25}{e}^{\mathrm{-5}x},$ $y(0)=$ $\mathrm{-2},$ $y\prime (0)=$ $0$

In each of the following problems, 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.

$x^2{y}^{″}-$ $2y=$ $10x^2-$ $1,$ $y_1(x)=$ $x^2,$ $y_2(x)=$ $x^{\mathrm{-1}}$

A 200-g mass stretches a spring 5 cm. Find the equation of motion of the mass if it is released from rest from a position 10 cm below the equilibrium position. What is the frequency of this motion?

Express the function $x(t)=$ $\text{cos}\left(4t\right)+$ $4\text{sin}\left(4t\right)$ in the form $A\text{sin}\left(\omega t+\varphi \right).$ What is the frequency of motion? The amplitude?

A 2-kg mass is attached to a spring with spring constant 24 N/m. The system is then immersed in a medium imparting a damping force equal to 16 times the instantaneous velocity of the mass. Find the equation of motion if it is released from rest at a point 40 cm below equilibrium.

A 1-lb weight stretches a spring 6 in., and the system is attached to a dashpot that imparts a damping force equal to half the instantaneous velocity of the mass. Find the equation of motion if the mass is released from rest at a point 6 in. below equilibrium.

A 1-kg mass stretches a spring 49 cm. The system is immersed in a medium that imparts a damping force equal to four times the instantaneous velocity of the mass. Find the equation of motion if the mass is released from rest at a point 24 cm above equilibrium.

A mass of 2 kg is attached to a spring with constant 32 N/m and comes to rest in the equilibrium position. Beginning at time $t=0,$ an external force equal to$f(t)=$ $68e^{\mathrm{-2}t}\text{cos}\left(4t\right)$ is applied to the system. Find the equation of motion if there is no damping. What is the transient solution? What is the steady-state solution?

Find the charge on the capacitor in an RLC series circuit where $L=1\text{/}5$ H, $R=2\text{/}5\text{Ω},$ $C=1\text{/}2$ F, and $E(t)=50$ V. Assume the initial charge on the capacitor is 0 C and the initial current is 4 A.

A mass weighing 2 lb stretches a spring 2 ft. Find the equation of motion if the spring is released from 2 in. below the equilibrium position with an upward velocity of 8 ft/sec. What is the period and frequency of the motion?

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

A block has a mass of 5 kg and is attached to a vertical spring with a spring constant of 20 N/m. The block is released from the equilibrium position with a downward velocity of 10 m/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?

An 800-lb weight (25 slugs) is attached to a vertical spring with a spring constant of 226 lb/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 ft below its equilibrium position with a downward velocity of 41 ft/sec.

b. Graph the solution and determine whether the motion is overdamped, critically damped, or underdamped.

A 1-kg mass stretches a spring 61.25 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 m below its equilibrium position with an upward velocity of 10 m/sec.

b. Determine whether the motion is overdamped, critically damped, or underdamped.

A 64-lb weight is attached to a vertical spring with a spring constant of 4.625 lb/ft. The resistance in the spring-mass system is equal to the instantaneous velocity. The weight is set in motion from a position 1 ft below its equilibrium position with an upward velocity of 2 ft/sec. Is the mass above or below the equilibrium position at the end of $\pi$ sec? By what distance?

A mass that weighs 6 lb stretches a spring 3 in. The system is acted on by an external force of $8\text{sin}\left(4t\right)$ 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 RLC series circuit where $L=2$ H, $R=24\text{Ω},$ $C=0.005$ F, and $E(t)=$ $12\text{sin}10t$ V. Assume the initial charge on the capacitor is 0.001 C and the initial current is 0 A.

A series circuit consists of a device where $L=\frac{1}{2}$ H, $R=10\text{Ω},$ $C=\frac{1}{50}$ F, and $E(t)=250$ V. If the initial charge on the capacitor is 0 C and the initial current is 18 A, find the charge and current at time t.

Find a power series solution for the following differential equations.

a. $y^{\prime }+2xy=0$

b. $(x+1)y^{\prime }=3y$

Find a power series solution for the following differential equations.

$5y^{″}+y^{\prime }=0$

Find a power series solution for the following differential equations.

$y^{″}-y=0$

Find a power series solution for the following differential equations.

$y^{\prime }-2xy=0$

Find a power series solution for the following differential equations.

$y^{″}-x{y}^{\prime }-y=0$

Find a power series solution for the following differential equations.

$x^2{y}^{″}-$ $x{y}^{\prime }-$ $3y=$ $0$

Find a power series solution for the following differential equations.

$y^{″}-$ $2xy=$ $0,$ $y\left(0\right)=$ $1,$ $y^{\prime }\left(0\right)=$ $\mathrm{-3}$

True or False? Justify your answer with a proof or a counterexample.

If $y$ and $z$ are both solutions to $y^{″}+2y^{\prime }+y=0,$ then $y+z$ is also a solution.

True or False? Justify your answer with a proof or a counterexample.

$y=$ $e^x\text{cos}(3x)+$ $e^x\text{sin}(2x)$ is a solution to the second-order differential equation $y^{″}+$ $2y\prime +$ $10=$ $0.$

Classify the differential equation. Determine the order, whether it is linear and, if linear, whether the differential equation is homogeneous or nonhomogeneous. If the equation is second-order homogeneous and linear, find the characteristic equation.

$y^{″}-2y=0$

Classify the differential equation. Determine the order, whether it is linear and, if linear, whether the differential equation is homogeneous or nonhomogeneous. If the equation is second-order homogeneous and linear, find the characteristic equation.

${\left(\frac{dy}{dt}\right)}^2+$ $y{y}^{\prime }=$ $1$

For the following problems, find the general solution.

$y^{″}+9y=0$

For the following problems, find the general solution.

$y^{″}-$ $2y^{\prime }+$ $10y=$ $4x$

For the following problems, find the general solution.

$y^{\text{'}\text{'}}+$ $5y\prime +$ $4y=$ $x+$ $e^{2x}$

For the following problems, find the general solution.

$y^{″}-$ $x^2=$ $\mathrm{-3}{y}^{\prime }-$ $\frac{9}{4}y+$ $3x$

For the following problems, find the solution to the initial-value problem, if possible.

$y^{″}+$ $4y^{\prime }+$ $6y=$ $0,$ $y\left(0\right)=0,$ $y^{\prime }\left(0\right)=\sqrt{2}$

For the following problems, find the solution to the boundary-value problem, if possible.

$4y^{\prime }=$ $\mathrm{-6}y+$ $2y^{″},$ $y\left(0\right)=0,$ $y\left(1\right)=1$

For the following problem, set up and solve the differential equation.

The motion of a swinging pendulum for small angles $\theta$ can be approximated by $\frac{d^2\theta }{d{t}^2}+$ $\frac{g}{L}\theta =$ $0,$ where $\theta$ is the angle the pendulum makes with respect to a vertical line, g is the acceleration resulting from gravity, and L is the length of the pendulum. Find the equation describing the angle of the pendulum at time $t,$ assuming an initial displacement of ${\theta }_0$ and an initial velocity of zero.

For the following problem, set up and solve the differential equations.

An opera singer is attempting to shatter a glass by singing a particular note. The vibrations of the glass can be modeled by $y^{″}+ay=0,$ where $a>0$ represents the natural frequency of the glass and the singer is forcing the vibrations at $\text{cos}(bt\text{).}$ For what value $b$ would the singer be able to break that glass? (Note: in order for the glass to break, the oscillations would need to get higher and higher.)