Chapter 3

In Problems 33-38, find a homogeneous Cauchy-Euler differential equation whose general solution is given. $$ y=c_{1} x^{-3}+c_{2} x^{-3} \ln x $$

$$ \text { In Problems 27-36, solve the given initial-value problem. } $$ $$ \begin{aligned} &y^{\prime \prime \prime}-2 y^{\prime \prime}+y^{\prime}=2-24 e^{x}+40 e^{5 x}, y(0)=\frac{1}{2}, y^{\prime}(0)=\frac{5}{2} \\ &y^{\prime \prime}(0)=-\frac{9}{2} \end{aligned} $$

(a) Verify that \(y_{p_{1}}=3 e^{2 x}\) and \(y_{p_{2}}=x^{2}+3 x\) are, respectively, particular solutions of $$ y^{\prime \prime}-6 y^{\prime}+5 y=-9 e^{2 x} $$ and \(\quad y^{\prime \prime}-6 y^{\prime}+5 y=5 x^{2}+3 x-16\). (b) Use part (a) to find particular solutions of $$ y^{\prime \prime}-6 y^{\prime}+5 y=5 x^{2}+3 x-16-9 e^{2 x} $$ and \(y^{\prime \prime}-6 y^{\prime}+5 y=-10 x^{2}-6 x+32+e^{2 x}\).

Show that the eigenvalues and eigenfunctions of the boundaryvalue problem $$ y^{\prime \prime}+\lambda y=0, \quad y(0)=0, \quad y(1)+y^{\prime}(1)=0 $$ are \(\lambda_{n}=\alpha_{n}^{2}\) and \(y_{n}=\sin \alpha_{n} x\), respectively, where \(\alpha_{n}, n=1\), \(2,3, \ldots\) are the consecutive positive roots of the equation \(\tan \alpha=-\alpha\)

$$ \text { Solve the given initial-value problem. } $$ $$ y^{\prime \prime \prime}+2 y^{\prime \prime}-5 y^{\prime}-6 y \quad 0, y(0) \quad y^{\prime}(0) \quad 0, y^{\prime \prime}(0) \quad 1 $$

(a) By inspection, find a particular solution of $$ y^{n}+2 y=10 $$ (b) By inspection, find a particular solution of $$ y^{\prime \prime}+2 y=-4 x $$ (c) Find a particular solution of \(y^{\prime \prime}+2 y=-4 x+10\). (d) Find a particular solution of \(y^{\prime \prime}+2 y=8 x+5\).

$$ y^{\prime \prime}+y=\sec ^{3} x, y(0)=1, y^{\prime}(0)=\frac{1}{2} $$

In Problems 33-38, find a homogeneous Cauchy-Euler differential equation whose general solution is given. $$ y=c_{1}+c_{2} x+c_{3} x \ln x $$

$$ \text { In Problems 27-36, solve the given initial-value problem. } $$ $$ y^{\prime \prime}+8 y=2 x-5+8 e^{-2 x}, y(0)=-5, y^{\prime}(0)=3, y^{\prime \prime}(0)=-4 $$

(a) By inspection, find a particular solution of $$ y^{\prime \prime}+2 y=10 . $$ (b) By inspection, find a particular solution of $$ y^{\prime \prime}+2 y=-4 x $$ (c) Find a particular solution of \(y^{\prime \prime}+2 y=-4 x+10\). (d) Find a particular solution of \(y^{\prime \prime}+2 y=8 x+5\).

In Problems \(37-40\), solve the given boundary-value problem. \(y^{\prime \prime}+y=x^{2}+1, y(0)=5, y(1)=0\)

Find a homogeneous Cauchy-Euler differential equation whose general solution is given. $$ y=c_{1} \cos (\ln x)+c_{2} \sin (\ln x) $$

Solve the given initial-value problem. $$ \frac{d^{2} x}{d t^{2}}+4 x=-5 \sin 2 t+3 \cos 2 t, x(0)=-1, x^{\prime}(0)=1 $$

In Problems 33-38, solve the given differential equation subject to the indicated conditions. $$ y^{\prime} y^{\prime \prime}=4 x, y(1)=5, y^{\prime}(1)=2 $$

In Problems 37 and 38, solve the given initial-value problem. $$ \frac{d^{2} x}{d t^{2}}+4 x=-5 \sin 2 t+3 \cos 2 t, x(0)=-1, x^{\prime}(0)=1 $$

$$ \text { In Problems } 37-40 \text {, solve the given boundary-value problem. } $$ $$ y^{\prime \prime}+y=x^{2}+1, y(0)=5, y(1)=0 $$

Solve the given boundary-value problem. \(y^{\prime \prime}-2 y^{\prime}+2 y=2 x-2, y(0)=0, y(\pi)=\pi\)

$$ \text { solve the given boundary-value problem. } $$ $$ \begin{array}{llll} y^{\prime \prime}+4 y & 0, y(0) & 0, y(\pi) & 0 \end{array} $$

Solve the given initial-value problem. $$ \frac{d^{2} x}{d t^{2}}+9 x=5 \sin 3 t, x(0)=2, x^{\prime}(0)=0 $$

Suppose that \(y_{1}=e^{x}\) and \(y_{2}=e^{-x}\) are two solutions of a homogeneous linear differential equation. Explain why \(y_{3}=\cosh x\) and \(y_{4}=\sinh x\) are also solutions of the equation.

In Problems 33-38, solve the given differential equation subject to the indicated conditions. $$ 2 y^{\prime \prime}=3 y^{2}, y(0)=1, y^{\prime}(0)=1 $$

In Problems 37 and 38, solve the given initial-value problem. $$ \frac{d^{2} x}{d t^{2}}+9 x=5 \sin 3 t, x(0)=2, x^{\prime}(0)=0 $$

In Problems 33-38, find a homogeneous Cauchy-Euler differential equation whose general solution is given. $$ y=c_{1} x^{1 / 2} \cos \left(\frac{1}{2} \ln x\right)+c_{2} x^{1 / 2} \sin \left(\frac{1}{2} \ln x\right) $$

$$ \text { In Problems } 37-40 \text {, solve the given boundary-value problem. } $$ $$ y^{\prime \prime}-2 y^{\prime}+2 y=2 x-2, y(0)=0, y(\pi)=\pi $$

In Problems , find the eigenvalues and eigenfunctions of the given boundary- value problem. Use a CAS to approximate the first four eigenvalues \(\lambda_{1}, \lambda_{2}, \lambda_{3}\), and \(\lambda_{4}\). $$ y^{\prime \prime}+\lambda y=0, y(0)=0, y(1)-\frac{1}{2} y^{\prime}(1)=0 $$

(a) Show that the solution of the initial-value problem $$ \begin{aligned} &\frac{d^{2} x}{d t^{2}}+\omega^{2} x=F_{0} \cos \gamma t, x(0)=0, x^{\prime}(0)=0 \\ &\text { is } \quad x(t)=\frac{F_{0}}{\omega^{2}-\gamma^{2}}(\cos \gamma t-\cos \omega t) \end{aligned} $$ (b) Evaluate \(\lim _{\gamma \rightarrow \infty} \frac{F_{0}}{\omega^{2}-\gamma^{2}}(\cos \gamma t-\cos \omega t)\).

(a) Use a CAS as an aid in finding the roots of the auxiliary equation for \(12 y^{(4)}+64 y^{\prime \prime \prime}+59 y^{\prime \prime}-23 y^{\prime}-12 y=0\). Give the general solution of the equation. (b) Solve the DE in part (a) subject to the initial conditions \(y(0)=-1, y^{\prime}(0)=2, y^{\prime \prime}(0)=5, y^{\prime \prime \prime}(0)=0\). Usea CASas an aid in solving the resulting systems of four equations in four unlenowns.

In Problems 39 and 40 , find the eigenvalues and eigenfunctions of the given boundary-value problem. Use a CAS to approximate the first four eigenvalues \(\lambda_{1}, \lambda_{2}, \lambda_{3}\), and \(\lambda_{4}\). $$ y^{\prime \prime}+\lambda y=0, y(0)=0, y(1)-\frac{1}{2} y^{\prime}(1)=0 $$

In Problems 39-42, use the substitution \(y=\left(x-x_{0}\right)^{m}\) to solve the given equation. $$ (x+3)^{2} y^{\prime \prime}-8(x+3) y^{\prime}+14 y=0 $$

$$ \text { In Problems } 37-40 \text {, solve the given boundary-value problem. } $$ $$ y^{\prime \prime}+3 y=6 x, y(0)=0, y(1)+y^{\prime}(1)=0 $$

Solve the given boundary-value problem. \(y^{\prime \prime}+3 y=6 x, y(0)+y^{\prime}(0)=0, y(1)=0\)

Use the substitution \(y=\left(x-x_{0}\right)^{m}\) to solve the given equation. $$ (x-1)^{2} y^{\prime \prime}-(x-1) y^{\prime}+5 y=0 $$

Is the set of functions \(f_{1}(x)=e^{x+2}, f_{2}(x)=e^{x-3}\) linearly dependent or linearly independent on the interval \((-\infty, \infty) ?\) Discuss.

In Problems 39 and 40 , find the eigenvalues and eigenfunctions of the given boundary-value problem. Use a CAS to approximate the first four eigenvalues \(\lambda_{1}, \lambda_{2}, \lambda_{3}\), and \(\lambda_{4}\). $$ y^{(4)}-\lambda y=0, y(0)=0, y^{\prime}(0)=0, y(1)=0, y^{\prime}(1)=0 $$

$$ (x-1)^{2} y^{\prime \prime}-(x-1) y^{\prime}+5 y=0 $$

$$ \text { In Problems } 37-40 \text {, solve the given boundary-value problem. } $$ $$ y^{\prime \prime}+3 y=6 x, y(0)+y^{\prime}(0)=0, y(1)=0 $$

In Problems 41 and 42, solve the given initial-value problem in which the input function \(g(x)\) is discontinuous. [Hint: Solve each problem on two intervals, and then find a solution so that \(y\) and \(y^{\prime}\) are continuous at \(x=\pi / 2\) (Problem 41 ) and at \(x=\pi\) (Problem 42).] \(y^{\prime \prime}+4 y=g(x), y(0)=1, y^{\prime}(0)=2\), where $$ g(x)=\left\\{\begin{array}{ll} \sin x, & 0 \leq x \leq \pi / 2 \\ 0, & x>\pi / 2 \end{array}\right. $$

Use the substitution \(y=\left(x-x_{0}\right)^{m}\) to solve the given equation. $$ (x+2)^{2} y^{\prime \prime}+(x+2) y^{\prime}+y=0 $$

Suppose \(y_{1}, y_{2}, \ldots, y_{k}\) are \(k\) linearly independent solutions on \((-\infty, \infty)\) of a homogeneous linear \(n\) th-order differential equation with constant coefficients. By Theorem \(3.1 .2\) it follows that \(y_{k+1}=0\) is also a solution of the differential equation. Is the set of solutions \(y_{1}, y_{2}, \ldots, y_{b} y_{k+1}\) linearly dependent or linearly independent on \((-\infty, \infty) ?\) Discuss.

In Problems \(41-44\), use systematic elimination to solve the given system. $$ \begin{aligned} &\frac{d x}{d t}+\frac{d y}{d t}=2 x+2 y+1 \\ &\frac{d x}{d t}+2 \frac{d y}{d t}=y+3 \end{aligned} $$

Solve the given initial-value problem in which the input function \(g(x)\) is discontinuous. [Hint: Solve each problem on two intervals, and then find a solution so that \(y\) and \(y^{\prime}\) are continuous at \(x=\pi / 2\) (Problem 41 ) and at \(x=\pi\) (Problem 42).] \(\begin{aligned} y^{\prime \prime}-2 y^{\prime}+10 y &=g(x), y(0)=0, y^{\prime}(0)=0, \text { where } \\ g(x) &=\left\\{\begin{array}{ll}20, & 0 \leq x \leq \pi \\ 0, & x>\pi\end{array}\right.\end{aligned}\)

Use the substitution \(y=\left(x-x_{0}\right)^{m}\) to solve the given equation. $$ (x-4)^{2} y^{\prime \prime}-5(x-4) y^{\prime}+9 y=0 $$

Suppose that \(y_{1}, y_{2}, \ldots, y_{k}\) are \(k\) nontrivial solutions of a homogeneous linear \(n\) th-order differential equation with constant coefficients and that \(k=n+1\). Is the set of solutions \(y_{1}, y_{2}, \ldots, y_{k}\) linearly dependent or linearly independent on \((-\infty, \infty) ?\) Discuss.

In Problems 39-42, use the substitution \(y=\left(x-x_{0}\right)^{m}\) to solve the given equation. $$ (x-4)^{2} y^{\prime \prime}-5(x-4) y^{\prime}+9 y=0 $$

In Problems 41 and 42 , solve the given initial-value problem in which the input function \(g(x)\) is discontinuous. \(y^{\prime \prime}-2 y^{\prime}+10 y=g(x), y(0)=0, y^{\prime}(0)=0\), where $$ g(x)= \begin{cases}20, & 0 \leq x \leq \pi \\ 0, & x>\pi\end{cases} $$

Consider the differental equation \(a y^{\prime \prime}+b y^{\prime}+c y=e^{k x}\), where \(a, b, c\), and \(k\) are constants. The auxiliary equation of the associated homogeneous equation is $$ a m^{2}+b m+c=0 $$ (a) If \(k\) is not a root of the auxiliary equation, show that we can find a particular solution of the form \(y_{p}=A e^{k x}\), where \(A=1 /\left(a k^{2}+b k+c\right)\) (b) If \(k\) is a root of the auxiliary equation of multiplicity one, show that we can find a particular solution of the form \(y_{p}=A x e^{k x}\), where \(A=1 /(2 a k+b) .\) Explain how we lenow that \(k \neq-b /(2 a)\) (c) If \(k\) is a root of the auxiliary equation of multiplicity two, show that we can find a particular solution of the form \(y=A x^{2} e^{k x}\), where \(A=1 /(2 a)\)

In Problems \(41-44\), use systematic elimination to solve the given system. $$ \begin{aligned} &(D-2) x \quad-y=-e^{t}\\\ &-3 x+(D-4) y=-7 e^{t} \end{aligned} $$

(a) Show that the general solution of $$ \frac{d^{2} x}{d t^{2}}+2 \lambda \frac{d x}{d t}+\omega^{2} x=F_{0} \sin \gamma t $$ is $$ \begin{aligned} x(t)=& A e^{-\lambda t} \sin \left(\sqrt{\omega^{2}-\lambda^{2} t}+\phi\right) \\\ &+\frac{F_{0}}{\sqrt{\left(\omega^{2}-\gamma^{2}\right)^{2}+4 \lambda^{2} \gamma^{2}}} \sin (\gamma t+\theta) \end{aligned} $$ where \(A=\sqrt{c_{1}^{2}+c_{2}^{2}}\) and the phase angles \(\phi\) and \(\theta\) are, respectively, defined by \(\sin \phi=c_{1} / A, \cos \phi=c_{2} / A\) and $$ \begin{aligned} &\sin \theta=\frac{-2 \lambda \gamma}{\sqrt{\left(\omega^{2}-\gamma^{2}\right)^{2}+4 \lambda^{2} \gamma^{2}}} \\ &\cos \theta=\frac{\omega^{2}-\gamma^{2}}{\sqrt{\left(\omega^{2}-\gamma^{2}\right)^{2}+4 \lambda^{2} \gamma^{2}}} \end{aligned} $$ (b) The solution in part (a) has the form \(x(t)=x_{c}(t)+x_{p}(t)\). Inspection shows that \(x_{e}(t)\) is ransient, and hence for large values of time, the solution is approximated by \(x_{p}(t)=\) \(g(\gamma) \sin (\gamma t+\theta)\), where $$ g(\gamma)=\frac{F_{0}}{\sqrt{\left(\omega^{2}-\gamma^{2}\right)^{2}+4 \lambda^{2} \gamma^{2}}} $$ Although the amplitude \(g(\gamma)\) of \(x_{p}(t)\) is bounded as \(t \rightarrow \infty\), show that the maximum oscillations will occur at the value \(\gamma_{1}=\sqrt{\omega^{2}-2 \lambda^{2}}\). What is the maximum value of \(g\) ? The number \(\sqrt{\omega^{2}-2 \lambda^{2}} / 2 \pi\) is said to be the resonance frequency of the system. (c) When \(F_{0}=2, m=1\), and \(k=4, g\) becomes $$ g(\gamma)=\frac{2}{\sqrt{\left(4-\gamma^{2}\right)^{2}+\beta^{2} \gamma^{2}}} $$ Construct a table of the values of \(\gamma_{1}\) and \(g\left(\gamma_{1}\right)\) corresponding to the damping coefficients \(\beta=2, \beta=1, \beta=\frac{3}{4}, \beta=\frac{1}{2}\), and \(\beta=\frac{1}{4}\). Use a graphing utility to obtain the graphs of \(g\) corresponding to these damping coefficients. Use the same coordinate axes. This family of graphs is called the resonance curve or frequency response curve of the system. What is \(\gamma_{1}\) approaching as \(\beta \rightarrow 0\) ? What is happening to the resonance curve as \(\beta \rightarrow 0\) ?

In Problems 43-48, use the substitution \(x=e^{t}\) to transform the given Cauchy-Euler equation to a differential equation with constant coefficients. Solve the original equation by solving the new equation using the procedures in Sections 3.3-3.5. $$ x^{2} y^{\prime \prime}+9 x y^{\prime}-20 y=0 $$

Use the substitution \(x=e^{t}\) to transform the given Cauchy-Euler equation to a differential equation with constant coefficients. Solve the original equation by solving the new equation using the procedures in Sections 3.3-3.5. $$ x^{2} y^{\prime \prime}-9 x y^{\prime}+25 y=0 $$