91Ó°ÊÓ

Consider the spring-mass system whose motion is governed by the initial-value problem \(\frac{d^{2} y}{d t^{2}}+3 \frac{d y}{d t}+2 y=0, \quad y(0)=1, \quad \frac{d y}{d t}(0)=-3\) (a) Determine the position of the mass at time \(t\) (b) Determine the time when the mass passes through the equilibrium position. (c) Make a sketch depicting the general motion of the system.

Show that the general solution for the motion of a critically damped spring- mass system, with initial displacement \(y_{0}\) and initial velocity \(v_{0},\) can be written in the form $$y(t)=e^{-c t /(2 m)}\left[y_{0}+t\left(v_{0}+\frac{c}{2 m} y_{0}\right)\right]$$ and that the system can pass through the equilibrium position at most once.

Solve the given differential equation on the interval \(x>0 .\) [Remember to put the equation in standard form.] $$x^{2} y^{\prime \prime}-4 x y^{\prime}+6 y=x^{4} \sin x$$

Determine a trial solution for the given nonhomogeneous differential equation. In each case, check that you obtain the same trial solution with or without the use of annihilators. $$y^{\prime \prime}+6 y^{\prime}+9 y=4 e^{-2 x}.$$

Determine three 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. $$y^{\prime \prime \prime}+y^{\prime \prime}-10 y^{\prime}+8 y=0$$

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. $$ y^{(i v)}-13 y^{\prime \prime}+36 y=0 $$

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.

Determine three 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 \((0, \infty)\). $$x^{3} y^{\prime \prime \prime}+3 x^{2} y^{\prime \prime}-6 x y^{\prime}=0, x > 0$$

Determine an appropriate trial solution for the given differential equation. Do not solve for the constants that arise in your trial solution. $$(D-2)(D-3) y=7 e^{2 x}$$.

Determine a particular solution to the given differential equation of the form $$ y_{p}(x)=A_{0}+A_{1} x+A_{2} x^{2} $$ Also find the general solution to the differential equation: $$ y^{\prime \prime}+y^{\prime}-2 y=4 x^{2}+5 $$