Chapter 3

The indicated functions are known linearly independent solutions of the associated homogeneous differential equation on the interval \((0, \infty)\). Find the general solution of the given nonhomogeneous equation. $$ \begin{aligned} &x^{2} y^{\prime \prime}+x y^{\prime}+\left(x^{2}-\frac{1}{4}\right) y=x^{3 / 2} ; \quad y_{1}=x^{-1 / 2} \cos x, \\ &y_{2}=x^{-1 / 2} \sin x \end{aligned} $$

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

A 1-kilogram mass is attached to a spring whose constant is \(16 \mathrm{~N} / \mathrm{m}\), and the entire system is then submerged in a liquid that imparts a darnping force numerically equal to 10 times the instantaneous velocity. Determine the equations of motion if (a) the mass is initially released from rest from a point 1 meter below the equilibrium position, and then (b) the mass is initially released from a point 1 meter below the equilibrium position with an upward velocity of \(12 \mathrm{~m} / \mathrm{s}\).

Verify that the given functions form a fundamental set of solutions of the differential equation on the indicated interval. Form the general solution. $$ y^{\prime \prime}-y^{\prime}-12 y=0 ; e^{-3 x}, e^{4 x},(-\infty, \infty) $$

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

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. $$ y^{\prime \prime}-y^{\prime}-12 y=0 ; e^{-3 x}, e^{4 x},(-\infty, \infty) $$

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

The indicated functions are known linearly independent solutions of the associated homogeneous differential equation on the interval \((0, \infty)\). Find the general solution of the given nonhomogeneous equation. $$ x^{2} y^{\prime \prime}+x y^{\prime}+y=\sec (\ln x) ; \quad y_{1}=\cos (\ln x), y_{2}=\sin (\ln x) $$

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

In Problems 23 and 24, the indicated functions are known linearly independent solutions of the associated homogeneous differential equation on the interval \((0, \infty)\). Find the general solution of the given nonhomogeneous equation. $$ x^{2} y^{\prime \prime}+x y^{\prime}+y=\sec (\ln x) ; \quad y_{1}=\cos (\ln x), y_{2}=\sin (\ln x) $$

In Problems 1-26, solve the given differential equation by undetermined coefficients. $$ y^{\prime \prime \prime}-y^{\prime \prime}-4 y^{\prime}+4 y=5-e^{x}+e^{2 x} $$

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. $$ y^{\prime \prime}-4 y=0 ; \cosh 2 x, \sinh 2 x,(-\infty, \infty) $$

Consider a pendulum that is released from rest from an initial displacement of \(\theta_{0}\) radians. Solving the linear model (7) subject to the initial conditions \(\theta(0)=\theta_{0}, \theta^{\prime}(0)=0\) gives \(\theta(t)=\theta_{0} \cos \sqrt{g} \| t .\) Theperiod of oscillations predicted by this modelisgivenbythefamiliarformula \(T=2 \pi / \sqrt{g l l}=2 \pi \sqrt{U g}\). The interesting thing about this formula for \(T\) is that it does not depend on the magnitude of the initial displacement \(\theta_{0}\). In other words, the linear model predicts that the time that it would take the pendulum to swing from an initial displacement of, say, \(\theta_{0}=\pi / 2\left(=90^{\circ}\right)\) to \(-\pi / 2\) and back again would be exactly the same time to cycle from, say, \(\theta_{0}=\pi / 360\left(=0.5^{\circ}\right)\) to \(-\pi / 360\). This is intuitively unreasonable; the actual period must depend on \(\theta_{0}\). If we assume that \(g=32 \mathrm{ft} / \mathrm{s}^{2}\) and \(l=32 \mathrm{ft}\), then the period of oscillation of the linear model is \(T=2 \pi \mathrm{s}\). Let us compare this last number with the period predicted by the nonlinear model when \(\theta_{0}=\pi / 4\). Using a numerical solver that is capable of generating hard data, approximate the solution of $$\frac{d^{2} \theta}{d t^{2}}+\sin \theta=0, \quad \theta(0)=\frac{\pi}{4}, \quad \theta^{\prime}(0)=0$$ for \(0 \leq t \leq 2\). As in Problem 24 , if \(t_{1}\) denotes the first time the pendulum reaches the position \(O P\) in Figure 3.11.3, then the period of the nonlinear pendulum is \(4 t_{1} .\) Here is another way of solving the equation \(\theta(t)=0\). Expeniment with small step sizes and advance the time staning at \(t=0\) and ending at \(t=2\). From your hard data, observe the time \(t_{1}\) when \(\theta(t)\) changes, for the first time, from positive to negative. Use the value \(t_{1}\) to determine the true value of the period of the nonlinear pendulum. Compute the percentage relative error in the period estimated by \(T=2 \pi\).

Solve the given differential equation by undetermined coefficients. \(y^{(4)}+2 y^{\prime \prime}+y=(x-1)^{2}\)

Solve the given initial-value problem. Use a graphing utility to graph the solution curve. $$ x^{2} y^{\prime \prime}+3 x y^{\prime}=0, y(1)=0, y^{\prime}(1)=4 $$

A force of 2 pounds stretches a spring 1 foot. A mass weighing \(3.2\) pounds is attached to the spring, and the system is then immersed in a medium that offers a damping force numerically equal to \(0.4\) times the instantaneous velocity. (a) Find the equation of motion if the mass is initially released from rest from a point 1 foot above the equilibrium position. (b) Express the equation of motion in the form given in (23). (c) Find the first time at which the mass passes through the equilibrium position heading upward.

In Problems \(15-28\), find the general solution of the given higher-order differential equation. $$ 16 \frac{d^{4} y}{d x^{4}}+24 \frac{d^{2} y}{d x^{2}}+9 y $$

Verify that the given functions form a fundamental set of solutions of the differential equation on the indicated interval. Form the general solution. $$ y^{\prime \prime}-2 y^{\prime}+5 y=0 ; e^{x} \cos 2 x, e^{x} \sin 2 x,(-\infty, \infty) $$

In Problems 25-30, solve the given initial-value problem. Use a graphing utility to graph the solution curve. $$ x^{2} y^{\prime \prime}+3 x y^{\prime}=0, y(1)=0, y^{\prime}(1)=4 $$

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

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. $$ y^{\prime \prime}-2 y^{\prime}+5 y=0 ; e^{x} \cos 2 x, e^{x} \sin 2 x,(-\infty, \infty) $$

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

Solve the given initial-value problem. Use a graphing utility to graph the solution curve. $$ x^{2} y^{\prime \prime}-5 x y^{\prime}+8 y=0, y(2)=32, y^{\prime}(2)=0 $$

In Problems \(15-28\), find the general solution of the given higher-order differential equation. $$ \frac{d^{4} y}{d x^{4}}-7 \frac{d^{2} y}{d x^{2}}-18 y $$

After a mass weighing 10 pounds is attached to a 5 -foot spring, the spring measures 7 feet. This mass is removed and replaced with another mass that weighs 8 pounds. The entire system is placed in a medium that offers a damping force numerically equal to the instantaneous velocity. (a) Find the equation of motion if the mass is initially released from a point \(\frac{1}{2}\) foot below the equilibrium position with a downward velocity of \(1 \mathrm{ft} / \mathrm{s}\). (b) Express the equation of motion in the form given in (23). (c) Find the times at which the mass passes through the equilibrium position heading downward. (d) Graph the equation of motion.

In Problems 25-30, solve the given initial-value problem. Use a graphing utility to graph the solution curve. $$ x^{2} y^{\prime \prime}-5 x y^{\prime}+8 y=0, y(2)=32, y^{\prime}(2)=0 $$

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

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. $$ 4 y^{\prime \prime}-4 y^{\prime}+y=0 ; e^{x / 2}, x e^{x / 2},(-\infty, \infty) $$

In Problems 27-36, solve the given initial-value problem. \(y^{\prime \prime}+4 y=-2, y(\pi / 8)=\frac{1}{2}, y^{\prime}(\pi / 8)=2\)

Crmsider the boundary-value problemintrodycod in theconstruction of the mathematical model for the shape of a rotating string: $$ T \frac{d^{2} y}{d x^{2}}+\rho \omega^{2} y=0, \quad y(0)=0, \quad y(L)=0 $$ For constant \(T\) and \(\rho\), define the critical speeds of angular rotation \(\omega_{n}\) as the values of \(\omega\) for which the boundary-value problem bas nontrivial solutions. Find the critical speeds \(\omega_{n}\) and the carrespanding deflections \(y_{n}(x)\).

Discuss how the methods of undetermined coefficients and variation of parameters can be combined to solve the given differential equation. Carry out your ideas. $$ 3 y^{\prime \prime}-6 y^{\prime}+30 y=15 \sin x+e^{x} \tan 3 x $$

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

Verify that the given functions form a fundamental set of solutions of the differential equation on the indicated interval. Form the general solution. $$ x^{2} y^{\prime \prime}-6 x y^{\prime}+12 y=0 ; x^{3}, x^{4},(0, \infty) $$

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

Consider the boundary-valueproblem introducod in theconstruction of the mathematical model for the shape of a rotating string: $$ T \frac{d^{2} y}{d x^{2}}+\rho w^{2} y=0, \quad y(0)=0, \quad y(L)=0 $$ For constant \(T\) and \(\rho\), define the mritical speeds of angular rotation \(\omega_{n}\) as the values of \(\omega\) for which the boundary-value pooblem has nontrivial solutions. Find the critical speeds \(\omega_{n}\) and the carrespanding deflections \(y_{n}(x)\).

A mass weighing 10 pounds stretches a spring 2 feet. The mass is attached to a dashpot damping device that offers a damping force numerically equal to \(\beta(\beta>0)\) times the instantaneous velocity. Determine the values of the damping constant \(\beta\) so that the subsequent motion is (a) overdamped, (b) critically damped, and (c) underdamped.

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

In Problems 27 and 28, discuss how the methods of undetermined coefficients and variation of parameters can be combined to solve the given differential equation. Carry out your ideas. $$ 3 y^{\prime \prime}-6 y^{\prime}+30 y=15 \sin x+e^{x} \tan 3 x $$

$$ \text { In Problems 27-36, solve the given initial-value problem. } $$ $$ y^{\prime \prime}+4 y=-2, \quad y(\pi / 8)=\frac{1}{2}, y^{\prime}(\pi / 8)=2 $$

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. $$ x^{2} y^{\prime \prime}-6 x y^{\prime}+12 y=0 ; x^{3}, x^{4},(0, \infty) $$

Solve the given initial-value problem. \(2 y^{\prime \prime}+3 y^{\prime}-2 y=14 x^{2}-4 x-11\) \(y(0)=0, y^{\prime}(0)=0\)

When the magnitude of tension \(T\) is not constant, then a model for the deflection curve or shape \(y(x)\) assumed by a rotating string is given by $$ \frac{d}{d x}\left[T(x) \frac{d y}{d x}\right]+\rho \omega^{2} y=0 $$ Suppose that \(10.25\), show that the critical speeds of angular rotation are $$ \omega_{n}=\frac{1}{2} \sqrt{\left(4 n^{2} \pi^{2}+1\right) / \rho} $$ and the corresponding deflections are $$ y_{n}(x)=c_{2} x^{-1 / 2} \sin (n \pi \ln x), n=1,2,3, \ldots $$ (b) Use a graphing utility to graph the deflection curves on the interval \([1, e]\) for \(n=1,2,3 .\) Choose \(c_{2}=1\)

Discuss how the methods of undetermined coefficients and variation of parameters can be combined to solve the given differential equation. Carry out your ideas. $$ y^{\prime \prime}-2 y^{\prime}+y=4 x^{2}-3+x^{-1} e^{x} $$

Solve the given initial-value problem. Use a graphing utility to graph the solution curve. $$ x^{2} y^{\prime \prime}-3 x y^{\prime}+4 y=0, y(1)=5, y^{\prime}(1)=3 $$

A mass weighing 24 pounds stretches a spring 4 feet. The subsequent motion takes place in a medium that offers a damping force numerically equal to \(\beta(\beta>0)\) times the instantaneous velocity. If the mass is initially released from the equilibrium position with an upward velocity of \(2 \mathrm{ft} / \mathrm{s}\), show that when \(\beta>3 \sqrt{2}\) the equation of motion is $$ x(t)=\frac{-3}{\sqrt{\beta^{2}-18}} e^{-2 \beta / 3} \sinh \frac{2}{3} \sqrt{\beta^{2}-18} t $$

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

In Problems 25-30, solve the given initial-value problem. Use a graphing utility to graph the solution curve. $$ x^{2} y^{\prime \prime}-3 x y^{\prime}+4 y=0, y(1)=5, y^{\prime}(1)=3 $$

$$ \text { In Problems 27-36, solve the given initial-value problem. } $$ $$ \begin{aligned} &2 y^{\prime \prime}+3 y^{\prime}-2 y=14 x^{2}-4 x-11 \\ &y(0)=0, y^{\prime}(0)=0 \end{aligned} $$

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. $$ x^{2} y^{\prime \prime}+x y^{\prime}+y=0 ; \cos (\ln x), \sin (\ln x),(0, \infty) $$

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