Chapter 3

A mass of 1 slug is suspended from a spring whose spring constant is \(9 \mathrm{lb} / \mathrm{ft}\). The mass is initially released from a point 1 foot above the equilibrium position with an upward velocity of \(\sqrt{3} \mathrm{ft} / \mathrm{s}\). Find the times for which the mass is heading downward at a velocity of \(3 \mathrm{ft} / \mathrm{s}\).

Solve the given differential equation. $$ x^{2} y^{\prime \prime}+8 x y^{\prime}+6 y=0 $$

Solve each differential equation by variation of parameters. $$ y^{\prime \prime}-2 y^{\prime}+y=\frac{e^{x}}{1+x^{2}} $$

FindaCauchy-Eulerdifferentialequation \(a x^{2} y^{\prime \prime}+b x y^{\prime}+c y=0\), where \(a, b\), and \(c\) are real constants, if it is known that (a) \(m_{1}=3\) and \(m_{2}=-1\) are roots of its auxiliary equation, (b) \(m_{1}=i\) is a complex root of its auxiliary equation.

In Problems \(1-20\), solve the given system of differential equations by systematic elimination. $$ \begin{aligned} &\left(2 D^{2}-D-1\right) x-(2 D+1) y=1\\\ &(D-1) x+\quad D y=-1 \end{aligned} $$

In Problems 11-20, find the eigenvalues and eigenfunctions for the given boundary-value problem. $$ y^{\prime \prime}+\lambda y=0, y^{\prime}(0)=0, y(\pi / 4)=0 $$

In Problems 1-18, solve the given differential equation. $$ x^{2} y^{\prime \prime}+8 x y^{\prime}+6 y=0 $$

In Problems \(1-18\), solve each differential equation by variation of parameters. $$ y^{\prime \prime}-2 y^{\prime}+y=\frac{e^{x}}{1+x^{2}} $$

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

In Problems \(1-16\), the indicated function \(y_{1}(x)\) is a solution of the given equation. Use reduction of order or formula (5), as instructed, to find a second solution \(y_{2}(x)\). $$ 4 x^{2} y^{\prime \prime}+y=0 ; \quad y_{1}=x^{1 / 2} \ln x $$

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

Solve the given differential equation by undetermined coefficients. \(y^{\prime \prime}+4 y=3 \sin 2 x\)

Solve the given system of differential equations by systematic elimination. $$ \begin{aligned} &2 \frac{d x}{d t}-5 x+\frac{d y}{d t}=e^{t} \\ &\frac{d x}{d t}-x+\frac{d y}{d t}=5 e^{t} \end{aligned} $$

Solve the given differential equation. $$ 3 x^{2} y^{\prime \prime}+6 x y^{\prime}+y=0 $$

The given two-parameter family is a solution of the indicated differential equation on the interval \((-\infty, \infty)\). Determine whether a member of the family can be found that satisfies the boundary conditions. \(y=c_{1} e^{x} \cos x+c_{2} e^{x} \sin x ; y^{\prime \prime}-2 y^{\prime}+2 y=0\) (a) \(y(0)=1, y^{\prime}(\pi)=0\) (b) \(y(0)=1, y(\pi)=-1\) (c) \(y(0)=1, y(\pi / 2)=1\) (d) \(y(0)=0, y(\pi)=0\)

In Problems 13-28, use the procedures developed in this chapter to find the general solution of each differential equation. $$ y^{\prime \prime}-2 y^{\prime}-2 y=0 $$

In Problems \(1-20\), solve the given system of differential equations by systematic elimination. $$ \begin{aligned} &2 \frac{d x}{d t}-5 x+\frac{d y}{d t}=e^{t} \\ &\frac{d x}{d t}-x+\frac{d y}{d t}=5 e^{t} \end{aligned} $$

In Problems 13-16, proceed as in Example 3 and obtain the first six nonzero terms of a Taylor series solution, centered at 0 , of the given initial-value problem. Use a numerical solver and a graphing utility to compare the solution curve with the graph of the Taylor polynomial. $$ y^{\prime \prime}=x+y^{2}, y(0)=1, y^{\prime}(0)=1 $$

In Problems 1-18, solve the given differential equation. $$ 3 x^{2} y^{\prime \prime}+6 x y^{\prime}+y=0 $$

In Problems \(1-18\), solve each differential equation by variation of parameters. $$ y^{\prime \prime}+3 y^{\prime}+2 y=\sin e^{x} $$

In Problems 1-26, solve the given differential equation by undetermined coefficients. $$ y^{\prime \prime}+4 y=3 \sin 2 x $$

In Problems \(1-16\), the indicated function \(y_{1}(x)\) is a solution of the given equation. Use reduction of order or formula (5), as instructed, to find a second solution \(y_{2}(x)\). $$ x^{2} y^{\prime \prime}-x y^{\prime}+2 y=0 ; \quad y_{1}=x \sin (\ln x) $$

In Problems 13 and 14, the given two-parameter family is a solution of the indicated differential equation on the interval \((-\infty, \infty)\). Determine whether a member of the family can be found that satisfies the boundary conditions. \(y=c_{1} e^{x} \cos x+c_{2} e^{x} \sin x ; y^{\prime \prime}-2 y^{\prime}+2 y=0\) (a) \(y(0)=1, y^{\prime}(\pi)=0\) (b) \(y(0)=1, y(\pi)=-1\) (c) \(y(0)=1, y(\pi / 2)=1\) (d) \(y(0)=0, y(\pi)=0\)

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

Solve the given differential equation by undetermined coefficients. \(y^{\prime \prime}-4 y=\left(x^{2}-3\right) \sin 2 x\)

Solve the given system of differential equations by systematic elimination. $$ \begin{aligned} &\frac{d x}{d t}+\frac{d y}{d t}=e^{t} \\ &-\frac{d^{2} x}{d t^{2}}+\frac{d x}{d t}+x+y=0 \end{aligned} $$

Solve the given differential equation. $$ x^{2} y^{\prime \prime}-7 x y^{\prime}+41 y=0 $$

The given two-parameter family is a solution of the indicated differential equation on the interval \((-\infty, \infty)\). Determine whether a member of the family can be found that satisfies the boundary conditions. \(y=c_{1} x^{2}+c_{2} x^{4}+3 ; x^{2} y^{\prime \prime}-5 x y^{\prime}+8 y=24\) (a) \(y(-1)=0, y(1)=4\) (b) \(y(0)=1, y(1)=2\) (c) \(y(0)=3, y(1)=0\) (d) \(y(1)=3, y(2)=15\)

In Problems 13-28, use the procedures developed in this chapter to find the general solution of each differential equation. $$ 2 y^{\prime \prime}+2 y^{\prime}+3 y=0 $$

In Problems \(1-20\), solve the given system of differential equations by systematic elimination. $$ \begin{aligned} &\frac{d x}{d t}+\frac{d y}{d t}=e^{t} \\ &-\frac{d^{2} x}{d t^{2}}+\frac{d x}{d t}+x+y=0 \end{aligned} $$

In Problems 11-20, find the eigenvalues and eigenfunctions for the given boundary-value problem. $$ y^{\prime \prime}+\lambda y=0, y(0)=0, \quad y^{\prime}(\pi / 2)=0 $$

In Problems 13-16, proceed as in Example 3 and obtain the first six nonzero terms of a Taylor series solution, centered at 0 , of the given initial-value problem. Use a numerical solver and a graphing utility to compare the solution curve with the graph of the Taylor polynomial. $$ y^{\prime \prime}+y^{2}=1, y(0)=2, y^{\prime}(0)=3 $$

In Problems 1-18, solve the given differential equation. $$ x^{2} y^{\prime \prime}-7 x y^{\prime}+41 y=0 $$

In Problems \(1-18\), solve each differential equation by variation of parameters. $$ y^{\prime \prime}-2 y^{\prime}+y=e^{t} \arctan t $$

In Problems 1-26, solve the given differential equation by undetermined coefficients. $$ y^{\prime \prime}-4 y=\left(x^{2}-3\right) \sin 2 x $$

In Problems \(1-16\), the indicated function \(y_{1}(x)\) is a solution of the given equation. Use reduction of order or formula (5), as instructed, to find a second solution \(y_{2}(x)\). $$ \begin{array}{ll} x^{2} y^{\prime \prime}-3 x y^{\prime}+5 y=0 ; & y_{1}=x^{2} \cos (\ln x) \end{array} $$

In Problems 13 and 14, the given two-parameter family is a solution of the indicated differential equation on the interval \((-\infty, \infty)\). Determine whether a member of the family can be found that satisfies the boundary conditions. $$ y=c_{1} x^{2}+c_{2} x^{4}+3 ; x^{2} y^{\prime \prime}-5 x y^{\prime}+8 y=24 $$ (a) \(y(-1)=0, y(1)=4\) (b) \(y(0)=1, y(1)=2\) (c) \(y(0)=3, y(1)=0\) (d) \(y(1)=3, y(2)=15\)

Solve the given differential equation by undetermined coefficients. \(y^{\prime \prime}+y=2 x \sin x\)

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

Solve the given system of differential equations by systematic elimination. $$ \begin{aligned} &(D-1) x+\left(D^{2}+1\right) y=1 \\ &\left(D^{2}-1\right) x+(D+1) y=2 \end{aligned} $$

Solve the given differential equation. $$ x^{3} y^{\prime \prime \prime}-6 y=0 $$

Solve each differential equation by variation of parameters. $$ y^{\prime \prime}+2 y^{\prime}+y=e^{-t} \ln t $$

Determine whether the given set of functions is linearly dependent or linearly independent on the interval \((-\infty, \infty)\). $$ f_{1}(x)=x, \quad f_{2}(x)=x^{2}, \quad f_{3}(x)=4 x-3 x^{2} $$

In Problems \(1-20\), solve the given system of differential equations by systematic elimination. $$ \begin{aligned} &(D-1) x+\left(D^{2}+1\right) y=1 \\ &\left(D^{2}-1\right) x+(D+1) y=2 \end{aligned} $$

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

In Problems 1-18, solve the given differential equation. $$ x^{3} y^{\prime \prime}-6 y=0 $$

In Problems \(1-18\), solve each differential equation by variation of parameters. $$ y^{\prime \prime}+2 y^{\prime}+y=e^{-t} \ln t $$

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

In Problems 15-22, determine whether the given set of functions is linearly dependent or linearly independent on the interval \((-\infty, \infty)\). $$ f_{1}(x)=x, \quad f_{2}(x)=x^{2}, \quad f_{3}(x)=4 x-3 x^{2} $$

Solve the given differential equation by undetermined coefficients. \(y^{\prime \prime}-5 y^{\prime}=2 x^{3}-4 x^{2}-x+6\)