Chapter 3

Solve the given initial-value problem. \(5 y^{\prime \prime}+y^{\prime}=-6 x, y(0)=0, y^{\prime}(0)=-10\)

Consider two concentric spheres of radius \(r=a\) and \(r=b, a

Solve the given initial-value problem. Use a graphing utility to graph the solution curve. $$ x y^{\prime \prime}+y^{\prime}=x, y(1)=1, y^{\prime}(1)=-\frac{1}{2} $$

A mass weighing 16 pounds stretches a spring \(\frac{8}{3}\) feet. The mass is initially released from rest from a point 2 feet below the equilibrium position, and the subsequent motion takes place in a medium that offers a damping force numerically equal to one-half the instantaneous velocity. Find the equation of motion if the mass is driven by an external force equal to \(f(t)=10 \cos 3 t\)

Verify that the given functions form a fundamental set of solutions of the differential equation on the indicated interval. Form the general solution. $$ x^{3} y^{\prime \prime \prime}+6 x^{2} y^{\prime \prime}+4 x y^{\prime}-4 y=0 ; x, x^{-2}, x^{-2} \ln x,(0, \infty) $$

Write down the form of the general solution \(y=y_{c}+y_{p}\) of the given differential equation in the two cases \(\omega \neq \alpha\) and \(\omega=\alpha\). Do not determine the coefficients in \(y_{p}\). (a) \(y^{\prime \prime}+\omega^{2} y=\sin \alpha x\) (b) \(y^{\prime \prime}-\omega^{2} y=e^{\alpha x}\)

Temperature in a Sphere Consider two concentric spheres of radius \(r=a\) and \(r=b, a

In Problems 25-30, solve the given initial-value problem. Use a graphing utility to graph the solution curve. $$ x y^{\prime \prime}+y^{\prime}=x, y(1)=1, y^{\prime}(1)=-\frac{1}{2} $$

$$ \text { In Problems 27-36, solve the given initial-value problem. } $$ $$ 5 y^{\prime \prime}+y^{\prime}=-6 x, y(0)=0, y^{\prime}(0)=-10 $$

In Problems \(23-30\), verify that the given functions form a fundamental set of solutions of the differential equation on the indicated interval. Form the general solution. $$ x^{3} y^{\prime \prime \prime}+6 x^{2} y^{\prime \prime}+4 x y^{\prime}-4 y=0 ; x, x^{-2}, x^{-2} \ln x,(0, \infty) $$

Solve the given initial-value problem. \(y^{\prime \prime}+4 y^{\prime}+4 y=(3+x) e^{-2 x}, y(0)=2, y^{\prime}(0)=5\)

Find the general solution of \(x^{4} y^{\prime \prime}+x^{3} y^{\prime}-4 x^{2} y=1\) given that \(y_{1}=x^{2}\) is a solution of the associated homogeneous equation.

Solve the given initial-value problem. Use a graphing utility to graph the solution curve. $$ x^{2} y^{\prime \prime}-5 x y^{\prime}+8 y=8 x^{6}, y\left(\frac{1}{2}\right)=0, y^{\prime}\left(\frac{1}{2}\right)=0 $$

A mass of 1 slug is attached to a spring whose constant is \(5 \mathrm{lb} / \mathrm{ft}\). Initially the mass is released 1 foot below the equilibrium position with a downward velocity of \(5 \mathrm{ft} / \mathrm{s}\), and the subsequent motion takes place in a medium that offers a damping force numerically equal to two times the instantaneous velocity. (a) Find the equation of motion if the mass is driven by an external force equal to \(f(t)=12 \cos 2 t+3 \sin 2 t\) (b) Graphthe transient and steady-state solutions on the same coordinate axes. (c) Graph the equation of motion.

Verify that the given functions form a fundamental set of solutions of the differential equation on the indicated interval. Form the general solution. $$ y^{(4)}+y^{\prime \prime}=0 ; 1, x, \cos x, \sin x,(-\infty, \infty) $$

(a) Given that \(y=\sin x\) is a solution of \(y^{(4)}+2 y^{\prime \prime \prime}+11 y^{\prime \prime}+\) \(2 y^{\prime}+10 y=0\), find the general solution of the DE without the aid of a calculator or a computer. (b) Find a linear second-order differential equation with constant coefficients for which \(y_{1}=1\) and \(y_{2}=e^{-x}\) are solutions of the associated homogeneous equation and \(y_{p}=\frac{1}{2} x^{2}-x\) is a particular solution of the nonhomogeneous equation.

In Problems 25-30, solve the given initial-value problem. Use a graphing utility to graph the solution curve. $$ x^{2} y^{\prime \prime}-5 x y^{\prime}+8 y=8 x^{6}, y\left(\frac{1}{2}\right)=0, y^{\prime}\left(\frac{1}{2}\right)=0 $$

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

In Problems \(23-30\), verify that the given functions form a fundamental set of solutions of the differential equation on the indicated interval. Form the general solution. $$ y^{(4)}+y^{\prime \prime}=0 ; 1, x, \cos x, \sin x,(-\infty, \infty) $$

Solve the given initial-value problem. \(y^{\prime \prime}+4 y^{\prime}+5 y=35 e^{-4 x}, y(0)=-3, y^{\prime}(0)=1\)

The indefinite integrals of the equations in (5) are nonelementary. Use a CAS to find the first four nonzero terms of a Maclaurin series of each integrand and then integrate the result. Find a particular solution of the given differential equation. $$ y^{\prime \prime}+y=\sqrt{1+x^{2}} $$

A mass of 1 slug, when attached to a spring, stretches it 2 feet and then comes to rest in the equilibrium position. Starting at \(t=0\), an external force equal to \(f(t)=8 \sin 4 t\) is applied to the system. Find the equation of motion if the surrounding medium offers a damping force numerically equal to eight times the instantaneous velocity.

Verify that the given two-parameter family of functions is the general solution of the nonhomogeneous differential equation on the indicated interval. $$ \begin{aligned} &y^{\prime \prime}-7 y^{\prime}+10 y=24 e^{x} \\ &y=c_{1} e^{2 x}+c_{2} e^{5 x}+6 e^{x},(-\infty, \infty) \end{aligned} $$

(a) Write the general solution of the fourth-order DE \(y^{(4)}-\) \(2 y^{\prime \prime}+y=0\) entirely in terms of hyperbolic functions. (b) Write down the form of a particular solution of \(y^{(4)}-\) \(2 y^{\prime \prime}+y=\sinh x\).

$$ \text { In Problems 27-36, solve the given initial-value problem. } $$ $$ y^{\prime \prime}+4 y^{\prime}+5 y=35 e^{-4 x}, y(0)=-3, y^{\prime}(0)=1 $$

In Problems 31-34, verify that the given two-parameter family of functions is the general solution of the nonhomogeneous differential equation on the indicated interval. $$ \begin{aligned} &y^{\prime \prime}-7 y^{\prime}+10 y=24 e^{x} \\ &y=c_{1} e^{2 x}+c_{2} e^{5 x}+6 e^{x},(-\infty, \infty) \end{aligned} $$

Solve the given initial-value problem. \(y^{\prime \prime}-y=\cosh x, y(0)=2, y^{\prime}(0)=12\)

The indefinite integrals of the equations in (5) are nonelementary. Use a CAS to find the first four nonzero terms of a Maclaurin series of each integrand and then integrate the result. Find a particular solution of the given differential equation. $$ 4 y^{\prime \prime}-y=e^{x^{2}} $$

$$ \text { Solve the given initial-value problem. } $$ $$ \begin{array}{llll} 4 y^{\prime \prime}-4 y^{\prime}-3 y & 0, y(0) & 1, y^{\prime}(0) & 5 \end{array} $$

Solve the given boundary-value problem. $$ x^{2} y^{\prime \prime}-3 x y^{\prime}+5 y=0, y(1)=0, y(e)=1 $$

Verify that the given two-parameter family of functions is the general solution of the nonhomogeneous differential equation on the indicated interval. $$ \begin{aligned} &y^{\prime \prime}+y=\sec x \\ &y=c_{1} \cos x+c_{2} \sin x+x \sin x+(\cos x) \ln (\cos x) \\ &(-\pi / 2, \pi / 2) \end{aligned} $$

Consider the differential equation \(x^{2} y^{\prime \prime}-\left(x^{2}+2 x\right) y^{\prime}+\) \((x+2) y=x^{3}\). Verify that \(y_{1}=x\) is one solution of the associated homogeneous equation. Then show that the method of reduction of order discussed in Section 3.2 leads both to a second solution \(y_{2}\) of the homogeneous equation and to a particular solution \(y_{p}\) of the nonhomogeneous equation. Form the general solution of the DE on the interval \((0, \infty)\).

In Problems 31 and 32, solve the given boundary-value problem. $$ x^{2} y^{\prime \prime}-3 x y^{\prime}+5 y=0, y(1)=0, y(e)=1 $$

$$ \text { In Problems 27-36, solve the given initial-value problem. } $$ $$ y^{\prime \prime}-y=\cosh x, \quad y(0)=2, y^{\prime}(0)=12 $$

In Problems 31-34, verify that the given two-parameter family of functions is the general solution of the nonhomogeneous differential equation on the indicated interval. $$ \begin{aligned} &y^{\prime \prime}+y=\sec x \\ &y=c_{1} \cos x+c_{2} \sin x+x \sin x+(\cos x) \ln (\cos x) \\ &(-\pi / 2, \pi / 2) \end{aligned} $$

When a mass of 2 leilograms is attached to a spring whose constant is \(32 \mathrm{~N} / \mathrm{m}\), it comes to rest in the equilibrium position. Starting at \(t=0\), a force equal to \(f(t)=68 e^{-2 t} \cos 4 t\) is applied to the system. Find the equation of motion in the absence of damping.

Verify that the given two-parameter family of functions is the general solution of the nonhomogeneous differential equation on the indicated interval. $$ \begin{aligned} &y^{\prime \prime}-4 y^{\prime}+4 y=2 e^{2 x}+4 x-12 \\ &y=c_{1} e^{2 x}+c_{2} x e^{2 x}+x^{2} e^{2 x}+x-2,(-\infty, \infty) \end{aligned} $$

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

When a mass of 2 lilograms is attached to a spring whose constant is \(32 \mathrm{~N} / \mathrm{m}\), it comes to rest in the equilibrium position. Starting at \(t=0\), a force equal to \(f(t)=68 e^{-2 t} \cos 4 t\) is applied to the system. Find the equation of motion in the absence of damping.

$$ \text { In Problems 27-36, solve the given initial-value problem. } $$ $$ \frac{d^{2} x}{d t^{2}}+\omega^{2} x=F_{0} \sin \omega t, x(0)=0, x^{\prime}(0)=0 $$

In Problems 31-34, verify that the given two-parameter family of functions is the general solution of the nonhomogeneous differential equation on the indicated interval. $$ \begin{aligned} &y^{\prime \prime}-4 y^{\prime}+4 y=2 e^{2 x}+4 x-12 \\ &y=c_{1} e^{2 x}+c_{2} x e^{2 x}+x^{2} e^{2 x}+x-2,(-\infty, \infty) \end{aligned} $$

Solve the given initial-value problem. \(\frac{d^{2} x}{d t^{2}}+\omega^{2} x=F_{0} \cos \gamma t, x(0)=0, x^{\prime}(0)=0\)

Verify that the given two-parameter family of functions is the general solution of the nonhomogeneous differential equation on the indicated interval. $$ \begin{aligned} &2 x^{2} y^{\prime \prime}+5 x y^{\prime}+y=x^{2}-x \\ &y=c_{1} x^{-1 / 2}+c_{2} x^{-1}+\frac{1}{15} x^{2}-\frac{1}{6} x,(0, \infty) \end{aligned} $$

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

$$ \text { In Problems 27-36, solve the given initial-value problem. } $$ $$ \frac{d^{2} x}{d t^{2}}+\omega^{2} x=F_{0} \cos \gamma t, x(0)=0, x^{\prime}(0)=0 $$

$$ \begin{aligned} &2 x^{2} y^{\prime \prime}+5 x y^{\prime}+y=x^{2}-x \\ &y=c_{1} x^{-1 / 2}+c_{2} x^{-1}+\frac{1}{15} x^{2}-\frac{1}{6} x,(0, \infty) \end{aligned} $$

Solve the given initial-value problem. \(y^{\prime \prime \prime}-2 y^{\prime \prime}+y^{\prime}=2-24 e^{x}+40 e^{5 x}, y(0)=\frac{1}{2}, y^{\prime}(0)=\frac{5}{2}\), \(y^{\prime \prime}(0)=-\frac{9}{2}\)

Consider the boundary-value problem $$ y^{\prime \prime}+\lambda y=0, y(-\pi)=y(\pi), y^{\prime}(-\pi)=y^{\prime}(\pi) $$ (a) The type of boundary conditions specified are called periodic boundary conditions. Give a geomeric interpretation of these conditions. (b) Find the eigenvalues and eigenfunctions of the problem. (c) Use a graphing utility to graph some of the eigenfunctions. Verify your geomeric interpretation of the boundary conditions given in part (a).

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

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