91Ó°ÊÓ

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

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

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.

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

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

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.

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. $$F(x)=e^{-3 x}(2 \sin x+7 \cos x)$$.

Find the solution to the Cauchy-Euler equation on the interval \((0, \infty) .\) In each case, \(m\) and \(k\) are positive constants. $$x^{2} y^{\prime \prime}-x(2 m-1) y^{\prime}+m^{2} y=0$$

Consider the spring-mass system whose motion is governed by the differential equation $$\frac{d^{2} y}{d t^{2}}+2 \alpha \frac{d y}{d t}+y=0$$ Determine all values of the (positive) constant \(\alpha\) for which the system is (i) underdamped, (ii) critically damped, and (iii) overdamped. In the case of overdamping, solve the system fully. If the initial velocity of the system is zero, determine if the mass passes through equilibrium.