Chapter 3

Determine whether the given set of functions is linearly dependent or linearly independent on the interval \((-\infty, \infty)\). $$ f_{1}(x)=x, \quad f_{2}(x)=x-1, \quad f_{3}(x)=x+3 $$

In Problems \(1-20\), solve the given system of differential equations by systematic elimination. $$ \begin{aligned} &\frac{d x}{d t}=6 y \\ &\frac{d y}{d t}=x+z \\ &\frac{d z}{d t}=x+y \end{aligned} $$

In Problems 11-20, find the eigenvalues and eigenfunctions for the given boundary-value problem. $$ x^{2} y^{\prime \prime}+x y^{\prime}+\lambda y=0, y(1)=0, y\left(e^{x}\right)=0 $$

In Problems 19-22, solve each differential equation by variation of parameters subject to the initial conditions \(y(0)=1, y^{\prime}(0)=0\) $$ 4 y^{\prime \prime}-y=x e^{x / 2} $$

In Problems 1-26, solve the given differential equation by undetermined coefficients. $$ y^{\prime \prime}+2 y^{\prime}+y=\sin x+3 \cos 2 x $$

In Problems 17-20, the indicated function \(y_{1}(x)\) is a solution of the associated homogeneous equation. Use the method of reduction of order to find a second solution \(y_{2}(x)\) of the homogeneous equation and a particular solution of the given nonhomogeneous equation. $$ y^{\prime \prime}-3 y^{\prime}+2 y=5 e^{3 x} ; \quad y_{1}=e^{x} $$

In Problems 15-22, determine whether the given set of functions is linearly dependent or linearly independent on the interval \((-\infty, \infty)\). $$ f_{1}(x)=x, \quad f_{2}(x)=x-1, \quad f_{3}(x)=x+3 $$

Solve the given differential equation by undetermined coefficients. \(y^{\prime \prime}+2 y^{\prime}-24 y=16-(x+2) e^{4 x}\)

In Problems, find the eigenvalues and eigenfunctions for the given boundary- value problem. $$ x^{2} y^{\prime \prime}+x y^{\prime}+\lambda y=0, y^{\prime}\left(e^{-1}\right)=0, y(1)=0 $$

Solve the given system of differential equations by systematic elimination. $$ \begin{aligned} &\frac{d x}{d t}=-x+z \\ &\frac{d y}{d t}=-y+z \\ &\frac{d z}{d t}=-x+y \end{aligned} $$

Discuss how to find an alternative two-parameter family of solutions for the nonlinear differential equation \(y^{\prime \prime}=2 x\left(y^{\prime}\right)^{2}\) in Example 1. [Hint: Suppose that \(-c_{1}^{2}\) is used as the constant of integration instead of \(\left.+c_{1}^{2} .\right]\)

Solve each differential equation by variation of parameters subject to the initial conditions \(y(0)=1, y^{\prime}(0)=0\). $$ 2 y^{\prime \prime}+y^{\prime}-y=x+1 $$

Determine whether the given set of functions is linearly dependent or linearly independent on the interval \((-\infty, \infty)\). $$ f_{1}(x)=2+x, \quad f_{2}(x)=2+|x| $$

In Problems 13-28, use the procedures developed in this chapter to find the general solution of each differential equation. $$ y^{\prime \prime}-2 y^{\prime}+y=x^{2} e^{x} $$

In Problems \(1-20\), solve the given system of differential equations by systematic elimination. $$ \begin{aligned} &\frac{d x}{d t}=-x+z \\ &\frac{d y}{d t}=-y+z \\ &\frac{d z}{d t}=-x+y \end{aligned} $$

In Problems 11-20, find the eigenvalues and eigenfunctions for the given boundary-value problem. $$ x^{2} y^{\prime \prime}+x y^{\prime}+\lambda y=0, y^{\prime}\left(e^{-1}\right)=0, y(1)=0 $$

In Problems 19-24, solve the given differential equation by variation of parameters. $$ 2 x^{2} y^{\prime \prime}+5 x y^{\prime}+y=x^{2}-x $$

In Problems 19-22, solve each differential equation by variation of parameters subject to the initial conditions \(y(0)=1, y^{\prime}(0)=0\) $$ 2 y^{\prime \prime}+y^{\prime}-y=x+1 $$

In Problems 1-26, solve the given differential equation by undetermined coefficients. $$ y^{\prime \prime}+2 y^{\prime}-24 y=16-(x+2) e^{4 x} $$

In Problems 17-20, the indicated function \(y_{1}(x)\) is a solution of the associated homogeneous equation. Use the method of reduction of order to find a second solution \(y_{2}(x)\) of the homogeneous equation and a particular solution of the given nonhomogeneous equation. $$ y^{\prime \prime}-4 y^{\prime}+3 y=x ; \quad y_{1}=e^{x} $$

Solve the given differential equation by undetermined coefficients. \(y^{\prime \prime \prime}-6 y^{\prime \prime}=3-\cos x\)

Discuss why the damping term in equation (3) is written as $$ \beta\left|\frac{d x}{d t}\right| \frac{d x}{d t} \text { instead of } \beta\left(\frac{d x}{d t}\right)^{2} $$

In Problems, find the eigenvalues and eigenfunctions for the given boundary- value problem. Consider only the case \(\lambda=\alpha^{4}, \alpha>0\) $$ \begin{aligned} &y^{(4)}-\lambda y=0, \quad y(0)=0, y^{\prime \prime}(0)=0, y(1)=0 \\ &y^{\prime \prime}(1)=0 \end{aligned} $$

Solve the given initial-value problem. $$ \begin{aligned} &\frac{d x}{d t}=-5 x-y \\ &\frac{d y}{d t}=4 x-y \\ &x(1)=0, y(1)=1 \end{aligned} $$

Motion in a Force Field A mathematical model for the position \(x(t)\) of a body moving rectilinearly on the \(x\) -axis in an inverse-square force field is given by $$ \frac{d^{2} x}{d t^{2}}=-\frac{k^{2}}{x^{2}} $$ Suppose that at \(t \quad 0\) the body starts from rest from the position \(x \quad x_{0}, x_{0}>0\). Show that the velocity of the body at time \(t\) is given by \(v^{2} \quad 2 k^{2}\left(1 / x-1 / x_{0}\right)\). Usethelastexpression anda CAS to carry out the integration to express time \(t\) in terms of \(x\).

Solve each differential equation by variation of parameters subject to the initial conditions \(y(0)=1, y^{\prime}(0)=0\). $$ y^{\prime \prime}+2 y^{\prime}-8 y=2 e^{-2 x}-e^{-x} $$

A mass weighing 4 pounds is attached to a spring whose constant is \(2 \mathrm{lb} / \mathrm{ft}\). The medium offers a damping force that is numerically equal to the instantaneous velocity. The mass is initially released from a point 1 foot above the equilibrium position with a downward velocity of \(8 \mathrm{ft} / \mathrm{s}\). Determine the time at which the mass passes through the equilibrium position. Find the time at which the mass attains its extreme displacement from the equilibrium position. What is the position of the mass at this instant?

Determine whether the given set of functions is linearly dependent or linearly independent on the interval \((-\infty, \infty)\). $$ f_{1}(x)=1+x, \quad f_{2}(x)=x, \quad f_{3}(x)=x^{2} $$

In Problems 13-28, use the procedures developed in this chapter to find the general solution of each differential equation. $$ y^{\prime \prime \prime}-5 y^{\prime \prime}+6 y^{\prime}=8+2 \sin x $$

In Problems 21 and 22 , solve the given initial-value problem. $$ \begin{aligned} &\frac{d x}{d t}=-5 x-y \\ &\frac{d y}{d t}=4 x-y \\ &x(1)=0, y(1)=1 \end{aligned} $$

In Problems 21 and 22, find the eigenvalues and eigenfunctions for the given boundary-value problem. Consider only the case \(\lambda=\alpha^{4}, \alpha>0\) $$ \begin{aligned} &y^{(4)}-\lambda y=0, \quad y(0)=0, \quad y^{\prime \prime}(0)=0, \quad y(1)=0 \\\ &y^{\prime \prime}(1)=0 \end{aligned} $$

In Problems 19-22, solve each differential equation by variation of parameters subject to the initial conditions \(y(0)=1, y^{\prime}(0)=0\) $$ y^{\prime \prime}+2 y^{\prime}-8 y=2 e^{-2 x}-e^{-x} $$

In Problems 15-22, determine whether the given set of functions is linearly dependent or linearly independent on the interval \((-\infty, \infty)\). $$ f_{1}(x)=1+x, \quad f_{2}(x)=x, \quad f_{3}(x)=x^{2} $$

Solve the given differential equation by undetermined coefficients. \(y^{\prime \prime \prime}-2 y^{\prime \prime}-4 y^{\prime}+8 y=6 x e^{2 x}\)

(a) Experiment with acalculator to find an interval \(0 \leq \theta<\theta_{1}\), where \(\theta\) is measured in radians, for which you think \(\sin \theta \approx \theta\) is a fairly good estimate. Then use a graphing utility to plot the graphs of \(y=x\) and \(y=\sin x\) on the same coordinate axes for \(0 \leq x \leq \pi / 2\). Do the graphs confirm your observations with the calculator? (b) Use a numerical solver to plot the solutions curves of the initial-value problems $$ \begin{aligned} \quad \frac{d^{2} \theta}{d t^{2}}+\sin \theta=0, & \theta(0)=\theta_{0}, \theta^{\prime}(0)=0 \\ \text { and } \quad \frac{d^{2} \theta}{d t^{2}}+\theta=0, \quad \theta(0)=\theta_{0}, \theta^{\prime}(0)=0 \end{aligned} $$ for several values of \(\theta_{0}\) in the interval \(0 \leq \theta<\theta_{1}\) found in part (a). Then plot solution curves of the initialvalue problems for several values of \(\theta_{0}\) for which \(\theta_{0}>\theta_{1}\)

In Problems, find the eigenvalues and eigenfunctions for the given boundary- value problem. Consider only the case \(\lambda=\alpha^{4}, \alpha>0\) $$ \begin{aligned} &y^{(4)}-\lambda y=0, \quad y^{\prime}(0)=0, y^{m \prime}(0)=0, y(x)=0, \\ &y^{\prime \prime}(\pi)=0 \end{aligned} $$

Solve the given initial-value problem. $$ \begin{aligned} &\frac{d x}{d t}=y-1 \\ &\frac{d y}{d t}=-3 x+2 y \\ &x(0)=0, y(0)=0 \end{aligned} $$

Solve each differential equation by variation of parameters subject to the initial conditions \(y(0)=1, y^{\prime}(0)=0\). $$ y^{\prime \prime}-4 y^{\prime}+4 y=\left(12 x^{2}-6 x\right) e^{2 x} $$

A 4 -foot spring measures 8 feet long after a mass weighing 8 pounds is attached to it. The medium through which the mass moves offers damping force numerically equal to \(\sqrt{2}\) times the instantaneous velocity. Find the equation of motion if the mass is initially released from the equilibrium position with a downward velocity of \(5 \mathrm{ft} / \mathrm{s}\). Find the time at which the mass attains its extreme displacement from the equilibrium position. What is the position of the mass at this instant?

Solve the given differential equation by variation of parameters. $$ x^{2} y^{\prime \prime}-2 x y^{\prime}+2 y=x^{4} e^{x} $$

Determine whether the given set of functions is linearly dependent or linearly independent on the interval \((-\infty, \infty)\). $$ f_{1}(x)=e^{x}, \quad f_{2}(x)=e^{-x}, \quad f_{3}(x)=\sinh x $$

In Problems 13-28, use the procedures developed in this chapter to find the general solution of each differential equation. $$ y^{\prime \prime \prime}-y^{\prime \prime}=6 $$

In Problems 21 and 22 , solve the given initial-value problem. $$ \begin{aligned} &\frac{d x}{d t}=y-1 \\ &\frac{d y}{d t}=-3 x+2 y \\ &x(0)=0, y(0)=0 \end{aligned} $$

In Problems 21 and 22, find the eigenvalues and eigenfunctions for the given boundary-value problem. Consider only the case \(\lambda=\alpha^{4}, \alpha>0\) $$ \begin{aligned} &y^{(4)}-\lambda y=0, \quad y^{\prime}(0)=0, \quad y^{m \prime}(0)=0, \quad y(\pi)=0 \\ &y^{\prime \prime}(\pi)=0 \end{aligned} $$

In Problems 19-24, solve the given differential equation by variation of parameters. $$ x^{2} y^{\prime \prime}-2 x y^{\prime}+2 y=x^{4} e^{x} $$

In Problems 19-22, solve each differential equation by variation of parameters subject to the initial conditions \(y(0)=1, y^{\prime}(0)=0\) $$ y^{\prime \prime}-4 y^{\prime}+4 y=\left(12 x^{2}-6 x\right) e^{2 x} $$

In Problems 1-26, solve the given differential equation by undetermined coefficients. $$ y^{\prime \prime \prime}-2 y^{\prime \prime}-4 y^{\prime}+8 y=6 x e^{2 x} $$

Verify that \(y_{1}(x)=x\) is a solution of \(x y^{\prime \prime}-x y^{\prime}+y=0\). Use reduction of order to find a second solution \(y_{2}(x)\) in the form of an infinite series. Conjecture an interval of definition for \(y_{2}(x)\).

In Problems 15-22, determine whether the given set of functions is linearly dependent or linearly independent on the interval \((-\infty, \infty)\). $$ f_{1}(x)=e^{x}, \quad f_{2}(x)=e^{-x}, \quad f_{3}(x)=\sinh x $$

Solve the given differential equation by undetermined coefficients. \(y^{\prime \prime \prime}-3 y^{\prime \prime}+3 y^{\prime}-y=x-4 e^{x}\)