91Ó°ÊÓ

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. $$\frac{d^{2} y}{d t^{2}}+2 \frac{d y}{d t}+5 y=0, \quad y(0)=1, \quad \frac{d y}{d t}(0)=3$$

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

Determine the current in the general RLC circuit with \(R^{2} < 4 L / C,\) if \(E(t)=E_{0} e^{-a t},\) where \(E_{0}, a\) are constants.

Determine the general solution to the given differential equation on \((0, \infty)\) $$x^{2} y^{\prime \prime}+x y^{\prime}+16 y=0$$

Use the variation-of-parameters method to find the general solution to the given differential equation. $$y^{\prime \prime}+9 y=\frac{36}{4-\cos ^{2}(3 x)}$$

Determine the annihilator of the given function. $$F(x)=(1-3 x) e^{4 x}+2 x^{2}$$.

Verify that the given function is in the kernel of \(L\). $$y(x)=\sin \left(x^{2}\right), \quad L=D^{2}-x^{-1} D+4 x^{2}$$

Determine a basis for the solution space to \(y^{\prime \prime}+y=0\) that is orthonormal on the interval \([-\pi, \pi].\)

Determine the annihilator of the given function. $$F(x)=e^{5 x}\left(2-x^{2}\right) \cos x$$.

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. $$\frac{d^{2} y}{d t^{2}}+3 \frac{d y}{d t}+2 y=0, \quad y(0)=1, \quad \frac{d y}{d t}(0)=0$$