Chapter 3

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

Solve the given system of differential equations by systematic elimination. $$ \begin{gathered} D^{2} x-2\left(D^{2}+D\right) y=\sin t \\ x+\quad D y=0 \end{gathered} $$

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

Solve each differential equation by variation of parameters. $$ 2 y^{\prime \prime}+2 y^{\prime}+y=4 \sqrt{x} $$

Determine whether the given set of functions is linearly dependent or linearly independent on the interval \((-\infty, \infty)\). $$ f_{1}(x)=0, \quad f_{2}(x)=x, \quad f_{3}(x)=e^{x} $$

In Problems 13-28, use the procedures developed in this chapter to find the general solution of each differential equation. $$ 2 y^{\prime \prime \prime}+9 y^{\prime \prime}+12 y^{\prime}+5 y=0 $$

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

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

A model of a spring/mass system is \(4 x^{\prime \prime}+i x=0\). By inspection of the differential equation only, discuss the behavior of the system over a long period of time.

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

In Problems \(1-18\), solve each differential equation by variation of parameters. $$ 2 y^{\prime \prime}+2 y^{\prime}+y=4 \sqrt{x} $$

In Problems 15-28, find the general solution of the given higher-order differential equation. $$ y^{\prime \prime \prime}-y \quad 0 $$

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 \(y_{2}(x)\). $$ \left(1-x^{2}\right) y^{\prime \prime}+2 x y^{\prime}=0 ; \quad y_{1}=1 $$

In Problems 15-22, determine whether the given set of functions is linearly dependent or linearly independent on the interval \((-\infty, \infty)\). $$ \begin{array}{lll} f_{1}(x)=0, & f_{2}(x)=x, & f_{3}(x)=e^{x} \end{array} $$

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

In Problems, find the eigenvalues and eigenfunctions for the given boundary- value problem. $$ y^{\prime \prime}+2 y^{\prime}+(\lambda+1) y=0, y(0)=0, y(5)=0 $$

Solve the given system of differential equations by systematic elimination. $$ \begin{aligned} &D x=y \\ &D y=z \\ &D z=x \end{aligned} $$

In calculus, the curvature of a curve that is defined by a function \(y=f(x)\) is defined as $$ \kappa=\frac{y^{\prime \prime}}{\left[1+\left(y^{\prime}\right)^{2}\right]^{3 / 2}} $$ Find \(y=f(x)\) for which \(\kappa=1 .\) [Hint: For simplicity, ignore constants of integration.]

Solve the given differential equation. $$ x y^{(4)}+6 y^{\prime \prime \prime}=0 $$

Determine whether the given set of functions is linearly dependent or linearly independent on the interval \((-\infty, \infty)\). $$ f_{1}(x)=5, \quad f_{2}(x)=\cos ^{2} x, \quad f_{3}(x)=\sin ^{2} x $$

In Problems 13-28, use the procedures developed in this chapter to find the general solution of each differential equation. $$ 3 y^{\prime \prime \prime}+10 y^{\prime \prime}+15 y^{\prime}+4 y=0 $$

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

In Problems 11-20, find the eigenvalues and eigenfunctions for the given boundary-value problem. $$ y^{\prime \prime}+2 y^{\prime}+(\lambda+1) y=0, y(0)=0, y(5)=0 $$

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. In calculus, the curvature of a curve that is defined by a function \(y=f(x)\) is defined as $$ \kappa=\frac{y^{\prime \prime}}{\left[1+\left(y^{\prime}\right)^{2}\right]^{3 / 2}} $$ Find \(y=f(x)\) for which \(\kappa=1\). [Hint: For simplicity, ignore constants of integration.]

In Problems 1-18, solve the given differential equation. $$ x y^{(4)}+6 y^{\prime \prime \prime}=0 $$

In Problems \(1-18\), solve each differential equation by variation of parameters. $$ 3 y^{\prime \prime}-6 y^{\prime}+6 y=e^{x} \sec x $$

In Problems 1-26, solve the given differential equation by undetermined coefficients. $$ y^{\prime \prime}-2 y^{\prime}+5 y=e^{x} \cos 2 x $$

In Problems 15-28, find the general solution of the given higher-order differential equation. $$ y^{\prime \prime \prime}-5 y^{\prime \prime}+3 y^{\prime}+9 y \quad 0 $$

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. $$ y^{\prime \prime}-4 y=2 ; \quad y_{1}=e^{-2 x} $$

In Problems 15-22, determine whether the given set of functions is linearly dependent or linearly independent on the interval \((-\infty, \infty)\). $$ f_{1}(x)=5, \quad f_{2}(x)=\cos ^{2} x, \quad f_{3}(x)=\sin ^{2} x $$

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

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

Solve the given system of differential equations by systematic elimination. $$ \begin{array}{r} D x+z=e^{t} \\ (D-1) x+D y+D z=0 \\ x+2 y+D z=e^{t} \end{array} $$

Solve each differential equation by variation of parameters. $$ 4 y^{\prime \prime}-4 y^{\prime}+y=e^{x / 2} \sqrt{1-x^{2}} $$

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

In Problems 13-28, use the procedures developed in this chapter to find the general solution of each differential equation. $$ 2 y^{(4)}+3 y^{\prime \prime \prime}+2 y^{\prime \prime}+6 y^{\prime}-4 y=0 $$

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

In Problems 13-18, proceed as in Example 3 to find the solution of the given initial-value problem. Evaluate the integral that defines \(y_{p}(x)\). $$ y^{\prime \prime}+y=\sec ^{2} x, \quad y(\pi)=0, y^{\prime}(\pi)=0 $$

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

In Problems 1-18, solve the given differential equation. $$ x^{4} y^{(4)}+6 x^{3} y^{\prime \prime \prime}+9 x^{2} y^{\prime \prime}+3 x y^{\prime}+y=0 $$

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

In Problems 15-28, find the general solution of the given higher-order differential equation. $$ y^{\prime \prime \prime}+3 y^{\prime \prime}-4 y^{\prime}-12 y \quad 0 $$

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. $$ y^{\prime \prime}+y^{\prime}=1 ; \quad y_{1}=1 $$

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

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

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

Solve the given system of differential equations by systematic elimination. $$ \begin{aligned} &\frac{d x}{d t}=6 y \\ &\frac{d y}{d t}=x+z \\ &\frac{d z}{d t}=x+y \end{aligned} $$

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

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

In Problems \(15-28\), find the general solution of the given higher-order differential equation. $$ \frac{d^{3} u}{d t^{3}}+\frac{d^{2} u}{d t^{2}}-2 u \quad 0 $$