91Ó°ÊÓ

A $\frac{1}{4}$– kg mass is attached to a spring with stiffness 8 N/m. The damping constant for the system is $\frac{1}{4}$N-sec/m. If the mass is moved 1 m to the left of equilibrium and released,what is the maximum displacement to the right that it will attain?

A mass weighing 8 lb is attached to a spring hanging from the ceiling and comes to rest at its equilibrium position. At t = 0, an external force F(t) = 2 cos 2t lb is applied to the system. If the spring constant is 10 lb/ft and the damping constant is 1 lb-sec/ft, find the equation of motion of the mass. What is the resonance frequency for the system?

A 2 kg mass is attached to a spring hanging from the ceiling, thereby causing the spring to stretch 20 cm upon coming to rest at equilibrium. At time t = 0, the mass is displaced 5 cm below the equilibrium position and released. At this same instant, an external force F(t) = 0.3 cos t N is applied to the systems. If the damping constant for the system is 5 N-sec/m, determine the equation of the motion for the mass. What is the resonance frequency for the system?

An 8-kg mass is attached to a spring hanging from the ceiling and allowed to come to rest. Assume that the spring constant is 40 N/m and the damping constant is 3 N/sec. At time t = 0, an external force $2\sin \left(2\mathrm{t}+\frac{\mathrm{Ï€}}{4}\right)$N is applied to the system. Determine the amplitude and frequency of the steady-state solution.

Decide whether or not the method of undetermined coefficients can be applied to find a particular solution to the given equation. $\mathbf{y}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{+}\mathbf{2}\mathbf{y}\mathbf{\text{'}}\mathbf{-}\mathbf{y}\mathbf{=}{\mathbf{t}}^{-1}{\mathbf{e}}^t$

In Problems 13–20, solve the given initial value problem.

y" - 4y' + 4y = 0 : y(1) = 1, y'(1) =1

Find a particular solution to the differential equation.

$\mathbf{y}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{+}\mathbf{2}\mathbf{y}\mathbf{\text{'}}\mathbf{+}\mathbf{2}\mathbf{y}\mathbf{=}\mathbf{4}{\mathbf{te}}^{-t}\mathbf{cost}$

If ${\mathit{F}}_{\mathit{e}\mathit{x}\mathit{t}}\mathbf{\left(}\mathit{t}\mathbf{\right)}\mathbf{=}\mathbf{0}$, equation (3) becomes $\mathit{m}\mathit{y}\text{'}\text{'}\mathbf{+}\mathit{b}\mathit{y}\text{'}\mathbf{+}\mathit{k}\mathit{y}\mathbf{=}\mathbf{0}$. For this equation, verify the following:

(a) If $\mathit{y}\mathbf{\left(}\mathit{t}\mathbf{\right)}$is a solution so is $\mathit{c}\mathit{y}\mathbf{\left(}\mathit{t}\mathbf{\right)}$, for any constant c.

(b) If${\mathit{y}}_{\mathbf{1}}\mathbf{\left(}\mathit{t}\mathbf{\right)}$and${\mathit{y}}_{\mathbf{2}}\mathbf{\left(}\mathit{t}\mathbf{\right)}$are solutions, so is their sum ${\mathit{y}}_{\mathbf{1}}\mathbf{\left(}\mathit{t}\mathbf{\right)}\mathbf{+}{\mathit{y}}_{\mathbf{2}}\mathbf{\left(}\mathit{t}\mathbf{\right)}$.

The auxiliary equations for the following differential equations have repeated complex roots. Adapt the "repeated root" procedure of Section $\mathbf{4}\mathbf{.}\mathbf{2}$ to find their general solutions:

$\mathbf{\left(}\mathbf{a}\mathbf{\right)}\mathbf{y}\text{'}\text{'}\text{'}\text{'}\mathbf{+}\mathbf{2}\mathbf{y}\text{'}\text{'}\mathbf{+}\mathbf{y}\mathbf{=}\mathbf{0}$

$\mathbf{\left(}\mathbf{b}\mathbf{\right)}\mathbf{y}\text{'}\text{'}\text{'}\text{'}\mathbf{+}\mathbf{4}\mathbf{y}\text{'}\text{'}\text{'}\mathbf{+}\mathbf{12}\mathbf{y}\text{'}\text{'}\mathbf{+}\mathbf{16}\mathbf{y}\text{'}\mathbf{+}\mathbf{16}\mathbf{y}\mathbf{=}\mathbf{0}$

Find a general solution $\mathbf{u}\text{'}\text{'}\mathbf{+}\mathbf{7}\mathbf{u}\mathbf{=}\mathbf{0}$