91Ó°ÊÓ

Determine four linearly independent solutions to the given differential equation of the form \(y(x)=e^{r x},\) and thereby determine the general solution to the differential equation. $$ y^{(i v)}-13 y^{\prime \prime}+36 y=0 $$

Determine the general solution to the given differential equation. $$(D+3)(D-1)(D+5)^{3} y=0$$

Use the result of the preceding problem to determine a particular solution to the given differential equation. $$\left(D^{2}-4 D+13\right)(D-3) y=F(x)$$

Solve the given initial-value problem: $$y^{\prime \prime}+9 y=5 \cos 2 x, y(0)=2, y^{\prime}(0)=3$$.

Consider the spring-mass system whose motion is governed by the differential equation $$\frac{d^{2} y}{d t^{2}}+2 \frac{d y}{d t}+5 y=4 e^{-t} \cos 2 t$$ (a) Describe the variation with time of the applied external force. (b) Determine the motion of the mass. What happens as \(t \rightarrow \infty ? ?\)

Determine the general solution to the given differential equation. $$\left(D^{2}+9\right)^{3} y=0$$

Consider the spring-mass system whose motion is governed by the differential equation$$\frac{d^{2} y}{d t^{2}}+16 y=130 e^{-t} \cos t$$ Determine the resulting motion, and identify any transient and steady-state parts of your solution.

Solve the given initial-value problem: $$y^{\prime \prime}-y=9 x e^{2 x}, y(0)=0, y^{\prime}(0)=7$$.

State whether the annihilator method can be used to determine a particular solution to the given differential equation. If the technique cannot be used, state why not. If the technique can be used, then give an appropriate trial solution. $$y^{\prime \prime}+y=4 \cos 2 x+3 e^{x}.$$

Determine two linearly independent solutions to the given differential equation of the form \(y(x)=x^{r},\) and thereby determine the general solution to the differential equation on \((0, \infty)\). $$x^{2} y^{\prime \prime}+3 x y^{\prime}-8 y=0, x > 0$$