Chapter 4

Solve the given initial-value problem. $$\begin{aligned} &\frac{d x}{d t}=-5 x-y\\\ &\frac{d y}{d t}=4 x-y\\\ &x(1)=0, y(1)=1 \end{aligned}$$

Solve the given differential equation by undetermined coefficients.In Problems \(1-26\) solve the given differential equation by undetermined coefficients. $$y^{\prime \prime \prime}-6 y^{\prime \prime}=3-\cos x$$

Solve each differential equation by variation of parameters, subject to the initial conditions \(y(0)=1, y^{\prime}(0)=0\). $$y^{\prime \prime}+2 y^{\prime}-8 y=2 e^{-2 x}-e^{-x}$$

Find a linear differential operator that annihilates the given function. $$13 x+9 x^{2}-\sin 4 x$$

Find the general solution of the given higher order differential equation. $$y^{\prime \prime \prime}+3 y^{\prime \prime}+3 y^{\prime}+y=0$$

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

Solve the given initial-value problem. $$\begin{aligned} &\frac{d x}{d t}=y-1\\\ &\frac{d y}{d t}=-3 x+2 y\\\ &x(0)=0, y(0)=0 \end{aligned}$$

Solve the given differential equation by undetermined coefficients.In Problems \(1-26\) solve the given differential equation by undetermined coefficients. $$y^{\prime \prime \prime}-2 y^{\prime \prime}-4 y^{\prime}+8 y=6 x e^{2 x}$$

Solve the given differential equation by variation of parameters. $$x^{2} y^{\prime \prime}-2 x y^{\prime}+2 y=x^{4} e^{x}$$

Solve each differential equation by variation of parameters, subject to the initial conditions \(y(0)=1, y^{\prime}(0)=0\). $$y^{\prime \prime}-4 y^{\prime}+4 y=\left(12 x^{2}-6 x\right) e^{2 x}$$

Find a linear differential operator that annihilates the given function. $$8 x-\sin x+10 \sin 5 x$$

Find the general solution of the given higher order differential equation. $$y^{\prime \prime \prime}-6 y^{\prime \prime}+12 y^{\prime}-8 y=0$$

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

Solve the given differential equation by undetermined coefficients.In Problems \(1-26\) solve the given differential equation by undetermined coefficients. $$y^{\prime \prime \prime}-3 y^{\prime \prime}+3 y^{\prime}-y=x-4 e^{x}$$

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

Find a linear differential operator that annihilates the given function. $$e^{-x}+2 x e^{x}-x^{2} e^{x}$$

Find the general solution of the given higher order differential equation. $$y^{(4)}+y^{\prime \prime \prime}+y^{\prime \prime}=0$$

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

Discuss how to find an alternative two-parameter family of solutions for the nonlinear differential equation \(y^{\prime \prime}=2 x\left(y^{\prime}\right)^{2}\) in Example 1. [Hint: Suppose that \(-c_{1}^{2}\) is used as the constant of integration instead of \(\left.+c_{1}^{2} .\right]\)

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

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

Find a linear differential operator that annihilates the given function. $$\left(2-e^{x}\right)^{2}$$

Verify that \(y_{1}(x)=x\) is a solution of \(x y^{\prime \prime}-x y^{\prime}+y=0\) Use reduction of order to find a second solution \(y_{2}(x)\) in the form of an infinite series. Conjecture an interval of definition for \(y_{2}(x)\)

Find the general solution of the given higher order differential equation. $$y^{(4)}-2 y^{\prime \prime}+y=0$$

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

Motion in a Force Field \(\quad\) A mathematical model for the position \(x(t)\) of a body moving rectilinearly on the \(x\) -axis in an inverse-square force field is given by \\[\frac{d^{2} x}{d t^{2}}=-\frac{k^{2}}{x^{2}}\\] Suppose that at \(t=0\) the body starts from rest from the position \(x=x_{0}, x_{0}>0 .\) Show that the velocity of the body at time \(t\) is given by \(v^{2}=2 k^{2}\left(1 / x-1 / x_{0}\right) .\) Use the last expression and a CAS to carry out the integration to express time \(t\) in terms of \(x\)

Solve the given differential equation by undetermined coefficients.In Problems \(1-26\) 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. $$x^{2} y^{\prime \prime}+3 x y^{\prime}=0, \quad y(1)=0, y^{\prime}(1)=4$$

Find a linear differential operator that annihilates the given function. $$3+e^{x} \cos 2 x$$

Find the general solution of the given higher order differential equation. $$16 \frac{d^{4} y}{d x^{4}}+24 \frac{d^{2} y}{d x^{2}}+9 y=0$$

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

A mathematical model for the position \(x(t)\) of a moving object is \\[\frac{d^{2} x}{d t^{2}}+\sin x=0\\] Use a numerical solver to graphically investigate the solutions of the equation subject to \(x(0)=0, x^{\prime}(0)=x_{1}, x_{1} \geq 0 .\) Discuss the motion of the object for \(t \geq 0\) and for various choices of \(x_{1}\) Investigate the equation \\[\frac{d^{2} x}{d t^{2}}+\frac{d x}{d t}+\sin x=0\\] in the same manner. Give a possible physical interpretation of the \(d x / d t\) term.

Solve the given differential equation by undetermined coefficients.In Problems \(1-26\) 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. $$x^{2} y^{\prime \prime}-5 x y^{\prime}+8 y=0, \quad y(2)=32, y^{\prime}(2)=0$$

Find a linear differential operator that annihilates the given function. $$e^{-x} \sin x-e^{2 x} \cos x$$

Find the general solution of the given higher order differential equation. $$\frac{d^{4} y}{d x^{4}}-7 \frac{d^{2} y}{d x^{2}}-18 y=0$$

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

Solve the given initial-value problem. $$y^{\prime \prime}+4 y=-2, \quad y(\pi / 8)=\frac{1}{2}, y^{\prime}(\pi / 8)=2$$

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

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

Find linearly independent functions that are annihilated by the given differential operator. $$D^{5}$$

Find the general solution of the given higher order differential equation. $$\frac{d^{5} u}{d r^{5}}+5 \frac{d^{4} u}{d r^{4}}-2 \frac{d^{3} u}{d r^{3}}-10 \frac{d^{2} u}{d r^{2}}+\frac{d u}{d r}+5 u=0$$

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

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

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

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

Find the general solution of the given higher order differential equation. $$2 \frac{d^{5} x}{d s^{5}}-7 \frac{d^{4} x}{d s^{4}}+12 \frac{d^{3} x}{d s^{3}}+8 \frac{d^{2} x}{d s^{2}}=0$$

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

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

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