91Ó°ÊÓ

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) \(y(x)=3 \ln x,(c) y(x)=\) \(2 e^{3 x}+3 \ln x\). $$L=D^{2}-x^{2} D+x$$

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

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

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 \(E(t)=13 \cos 3 t \mathrm{V}\)

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

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

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

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 \(t \rightarrow+\infty ?\)