Q3E - Fundamentals Of Differential Equations And Boundary Value Problems

Question

Question: Show that if ${\mathbf{F}}_{\mathbf{ext}}\mathbf{\left(}\mathbf{t}\mathbf{\right)}\mathbf{=}\mathbf{0}\mathbf{,}\mathbf{m}\mathbf{=}\mathbf{1}\mathbf{,}\mathbf{k}\mathbf{=}\mathbf{9}$, and , then the equation   has the "critically damped" solutions ${\mathbf{y}}_{\mathbf{1}}\mathbf{\left(}\mathbf{t}\mathbf{\right)}\mathbf{=}{\mathbf{e}}^{\mathbf{-}\mathbf{3}\mathbf{t}}$ and ${\mathbf{y}}_{\mathbf{2}}\mathbf{\left(}\mathbf{t}\mathbf{\right)}\mathbf{=}\mathbf{t}{\mathbf{e}}^{\mathbf{-}\mathbf{3}\mathbf{t}}$ . What is the limit of these solutions as $\mathbf{t}\mathbf{\to }\mathbf{\infty }$?

 

Step-by-Step Solution

Answer

Answer

 

The $\mathbf{y}\mathbf{\left(}\mathbf{t}\mathbf{\right)}\mathbf{=}{\mathbf{c}}_{\mathbf{1}}{\mathbf{e}}^{\mathbf{-}\mathbf{3}\mathbf{t}}\mathbf{+}{\mathbf{c}}_{\mathbf{2}}{\mathbf{te}}^{\mathbf{-}\mathbf{3}\mathbf{t}}$ solution is and the limit of the solution is .

Step 1: Finding the differential equation and the general solution.

The differential equation for the mass-spring oscillator is 

$\mathbf{my}\text{'}\text{'}\mathbf{+}\mathbf{by}\text{'}\mathbf{+}\mathbf{ky}\mathbf{=}{\mathbf{F}}_{\mathbf{ext}}.$

Given ${\mathbf{F}}_{\mathbf{ext}}\mathbf{=}\mathbf{0}\mathbf{,}\mathbf{m}\mathbf{=}\mathbf{1}\mathbf{,}\mathbf{k}\mathbf{=}\mathbf{9}$ and , then the differential equation is $\mathbf{y}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{+}\mathbf{6}\mathbf{y}\mathbf{\text{'}}\mathbf{+}\mathbf{9}\mathbf{y}\mathbf{=}\mathbf{0}.$ 

 

The auxiliary equation for the given differential equation is,$$\begin{gathered} {\mathbf{m}}^{\mathbf{2}}\mathbf{+}\mathbf{6}\mathbf{m}\mathbf{+}\mathbf{9}\mathbf{=}\mathbf{0} \\ {\mathbf{(}\mathbf{m}\mathbf{+}\mathbf{3}\mathbf{)}}^{\mathbf{2}}\mathbf{=}\mathbf{0} \end{gathered}$$

 

 

Step 2: Check for critically damped.

If , then the system is critically damped. Here  and , then;

 $\begin{array}{rcl}{b}^2-4ac& =& {\mathbf{6}}^{\mathbf{2}}\mathbf{-}\mathbf{9}\mathbf{\times }\mathbf{4}\\ & \mathbf{=}& \mathbf{0}\\ & & \end{array}$

 

So, the system is critically damped and , then the solutions are; 

 $\mathbf{y}\mathbf{\left(}\mathbf{t}\mathbf{\right)}\mathbf{=}{\mathbf{c}}_{\mathbf{1}}{\mathbf{e}}^{\mathbf{-}\mathbf{3}\mathbf{t}}\mathbf{+}{\mathbf{c}}_{\mathbf{2}}{\mathbf{te}}^{\mathbf{-}\mathbf{3}\mathbf{t}}$

 

If $\mathbf{t}\to \infty$, then ${\mathbf{e}}^{\mathbf{-}\mathbf{3}\mathbf{t}}\mathbf{=}\mathbf{0}$. Therefore,