Chapter 8

Find \(L y\) for the given differential operator \(L\) and the given function \(y.\) $$L=D^{2}+3, \quad y(x)=e^{x^{3}}.$$

Determine the general solution to the given differential equation on \((0, \infty)\) $$x^{2} y^{\prime \prime}-4 x y^{\prime}+4 y=0$$

Determine the charge on the capacitor at time \(t\) in an RLC circuit that has \(R=4 \Omega, L=4 \mathrm{H}, C=\frac{1}{17} \mathrm{F}\) and \(E=E_{0} \mathrm{V},\) where \(E_{0}\) is constant. What happens to the charge on the capacitor as \(t \rightarrow+\infty ?\) Describe the behavior of the current in the circuit.

Use the variation-of-parameters method to find the general solution to the given differential equation. $$y^{\prime \prime}-6 y^{\prime}+9 y=4 e^{3 x} \ln x, \quad x>0$$

For Problems \(1-2,\) consider the spring-mass system whose motion is governed by the given initial-value problem. Determine the circular frequency of the system and the amplitude, phase, and period of the motion. $$\frac{d^{2} y}{d t^{2}}+4 y=0, \quad y(0)=2, \quad \frac{d y}{d t}(0)=4$$

Determine a basis for the solution space of the given differential equation. $$y^{\prime \prime}+2 y^{\prime}-3 y=0$$

Determine the annihilator of the given function. $$F(x)=5 e^{-3 x}$$.

Find \(L y\) for the given differential operator if \((a) y(x)=2 e^{3 x},\) (b) \(y(x)=3 \ln x,(c) y(x)=\) \(2 e^{3 x}+3 \ln x\). $$L=D-x$$

For all problems below, use a complex-valued trial solution to determine a particular solution to the given differential equation. $$y^{\prime \prime}-16 y=20 \cos 4 x$$

Determine the general solution to the given differential equation on \((0, \infty)\) $$x^{2} y^{\prime \prime}+3 x y^{\prime}+y=0$$

Determine the steady-state current in the RLC circuit that has \(R=\frac{3}{2} \Omega, L=\frac{1}{2} \mathrm{H}, C=\frac{2}{3} \mathrm{F},\) and \(E(t)=13 \cos 3 t \mathrm{V}\)

Use the variation-of-parameters method to find the general solution to the given differential equation. $$y^{\prime \prime}+4 y^{\prime}+4 y=x^{-2} e^{-2 x}, \quad x>0$$

Consider the spring-mass system whose motion is governed by the given initial- value problem. Determine the circular frequency of the system and the amplitude, phase, and period of the motion. \(\frac{d^{2} y}{d t^{2}}+\omega_{0}^{2} y=0, \quad y(0)=y_{0}, \frac{d y}{d t}(0)=v_{0}\) where \(\omega_{0}, y_{0}, v_{0}\) are constants.

Determine a basis for the solution space of the given differential equation. $$y^{\prime \prime}+6 y^{\prime}+9 y=0$$

Determine the annihilator of the given function. $$F(x)=2 e^{x}-3 x$$.

Find \(L y\) for the given differential operator if \((a) y(x)=2 e^{3 x},\) (b) \(y(x)=3 \ln x,(c) y(x)=\) \(2 e^{3 x}+3 \ln x\). $$L=D^{2}-x^{2} D+x$$

For all problems below, use a complex-valued trial solution to determine a particular solution to the given differential equation. $$y^{\prime \prime}+2 y^{\prime}+y=50 \sin 3 x$$

Determine the general solution to the given differential equation on \((0, \infty)\) $$x^{2} y^{\prime \prime}+5 x y^{\prime}+13 y=0$$

Consider the RLC circuit with \(E(t)=E_{0} \cos \omega t\) V, where \(E_{0}\) and \(\omega\) are constants. If there is no resistor in the circuit, show that the charge on the capacitor satisfies $$ \lim _{t \rightarrow \infty} q(t)=+\infty $$ if and only if \(\omega=\frac{1}{\sqrt{L C}} .\) What happens to the current in the circuit as \(t \rightarrow+\infty ?\)

Use the variation-of-parameters method to find the general solution to the given differential equation. $$y^{\prime \prime}+9 y=18 \sec ^{3}(3 x), \quad|x|<\pi / 6$$

A force of \(3 \mathrm{N}\) stretches a spring by \(1 \mathrm{m}\). (a) Find the spring constant \(k\) (b) A mass of \(4 \mathrm{kg}\) is attached to the spring. At \(t=0\) the mass is pulled down a distance 1 meter from equilibrium and released with a downward velocity of 0.5 meters/second. Assuming that damping is negligible, determine an expression for the position of the mass at time \(t .\) Find the circular frequency of the system and the amplitude, phase, and period of the motion.

Determine a basis for the solution space of the given differential equation. $$y^{\prime \prime}-6 y^{\prime}+25 y=0$$

Determine the annihilator of the given function. $$F(x)=\sin x+3 x e^{2 x}$$.

Find \(L y\) for the given differential operator if \((a) y(x)=2 e^{3 x},\) (b) \(y(x)=3 \ln x,(c) y(x)=\) \(2 e^{3 x}+3 \ln x\). $$L=D^{3}-2 x D^{2}$$

For all problems below, use a complex-valued trial solution to determine a particular solution to the given differential equation. $$y^{\prime \prime}-y=10 e^{2 x} \cos x$$

Find \(L y\) for the given differential operator \(L\) and the given function \(y.\) $$L=x^{2} D^{3}-\sin x D, \quad y(x)=e^{2 x}+\cos x.$$

Determine the general solution to the given differential equation on \((0, \infty)\) $$x^{2} y^{\prime \prime}-x y^{\prime}+5 y=0$$

Consider the RLC circuit with \(R=3 \Omega, L=\frac{1}{2} \mathrm{H}\) \(C=\frac{1}{5} \mathrm{F},\) and \(E(t)=2 \cos \omega t\) V. Determine the cur- rent in the circuit at time \(t,\) and find the value of \(\omega\) that maximizes the amplitude of the steady-state current.

Determine the annihilator of the given function. $$F(x)=x^{3} e^{7 x}+5 \sin 4 x$$.

Use the variation-of-parameters method to find the general solution to the given differential equation. $$y^{\prime \prime}+6 y^{\prime}+9 y=\frac{2 e^{-3 x}}{x^{2}+1}$$

Determine the motion of the spring-mass system governed by the given initial- value problem. In each case, state whether the motion is underdamped, critically damped, or overdamped, and make a sketch depicting the motion. $$\frac{d^{2} y}{d t^{2}}+2 \frac{d y}{d t}+y=0, \quad y(0)=-1, \quad \frac{d y}{d t}(0)=2$$

Let \(S\) denote the subspace of the solution space to the differential equation \(y^{\prime \prime}+9 y=0,\) with basis \(\\{2 \sin 3 x-7 \cos 3 x\\} .\) Write the general vector in \(S\) and extend the basis for \(S\) to a basis for the full solution space of the differential equation.

Find \(L y\) for the given differential operator if \((a) y(x)=2 e^{3 x},\) (b) \(y(x)=3 \ln x,(c) y(x)=\) \(2 e^{3 x}+3 \ln x\). $$L=D^{3}-D+4$$

For all problems below, use a complex-valued trial solution to determine a particular solution to the given differential equation. $$y^{\prime \prime}+4 y^{\prime}+4 y=169 \sin 3 x$$

Find \(L y\) for the given differential operator \(L\) and the given function \(y.\) $$L=\left(x^{2}+1\right) D^{3}-(\cos x) D+5 x^{2}, y(x)=\ln x+8 x^{5}.$$

Determine the general solution to the given differential equation on \((0, \infty)\) $$x^{2} y^{\prime \prime}-6 y=0$$

Consider the RLC circuit with \(R=16 \Omega, L=8 \mathrm{H}\) \(C=\frac{1}{40} \mathrm{F},\) and \(E(t)=17 \cos 2 t \mathrm{V} .\) Determine the current in the circuit for \(t > 0,\) given that at \(t=0,\) the capacitor is uncharged and there is no current flowing.

Verify that the given function is in the kernel of \(L\). $$y(x)=x e^{2 x}, \quad L=D^{2}-4 D+4$$

Determine the annihilator of the given function. $$F(x)=4 e^{-2 x} \sin x$$.

Use the variation-of-parameters method to find the general solution to the given differential equation. $$y^{\prime \prime}-4 y=\frac{8}{e^{2 x}+1}$$

Determine the motion of the spring-mass system governed by the given initial- value problem. In each case, state whether the motion is underdamped, critically damped, or overdamped, and make a sketch depicting the motion. $$4 \frac{d^{2} y}{d t^{2}}+4 \frac{d y}{d t}+y=0, \quad y(0)=4, \quad \frac{d y}{d t}(0)=-1$$

For problems \(5-7\) you will need to use the function inner product \(=\int_{a}^{b} f(x) g(x) d x\) in \(C^{2}[a, b].\) Determine a basis for the solution space to \(y^{\prime \prime}+y^{\prime}-\) \(2 y=0\) that is orthogonal on the interval [0,1].

For all problems below, use a complex-valued trial solution to determine a particular solution to the given differential equation. $$y^{\prime \prime}-y^{\prime}-2 y=40 \sin ^{2} x$$

Determine the annihilator of the given function. $$F(x)=e^{x} \sin 2 x+3 \cos 2 x$$.

Find \(L y\) for the given differential operator \(L\) and the given function \(y.\) $$L=4 x^{2 D}, \quad y(x)=\sin ^{2}\left(x^{2}+1\right).$$

Determine the general solution to the given differential equation on \((0, \infty)\) $$x^{2} y^{\prime \prime}-3 x y^{\prime}+4 y=0$$

Show that the differential equation governing the behavior of an RLC circuit can be written directly in terms of the current \(i(t)\) has $$ \frac{d^{2} i}{d t^{2}}+\frac{R}{L} \frac{d i}{d t}+\frac{1}{L C} i=\frac{1}{L} \frac{d E}{d t} $$

Use the variation-of-parameters method to find the general solution to the given differential equation. $$y^{\prime \prime}-4 y^{\prime}+5 y=e^{2 x} \tan x, \quad 0

For problems \(5-7\) you will need to use the function inner product \(=\int_{a}^{b} f(x) g(x) d x\) in \(C^{2}[a, b].\) Determine a basis for the solution space to \(y^{\prime \prime}+4 y=0\) that is orthogonal on the interval \([0, \pi / 4].\)

For all problems below, use a complex-valued trial solution to determine a particular solution to the given differential equation. $$y^{\prime \prime}+y=3 e^{x} \cos 2 x$$