Q2 E - Fundamentals Of Differential Equations And Boundary Value Problems

Question

Using the paradigm (13), what are the inertia, damping, and stiffness for the equation $\mathrm{y}-6{\mathrm{y}}^2=0$? If y > 0, what is the sign of the “stiffness constant”? Does your answer help explain the runaway behaviour of the solutions $\mathrm{y}\left(\mathrm{t}\right)=\frac{1}{{\left(\mathrm{c}-\mathrm{t}\right)}^2}$?

Step-by-Step Solution

Answer

The sign of the stiffness constant is negative. And, stiffness is $\mathrm{k}=-6\mathrm{y}$. Then, the answer helps in explaining the runaway behaviour of the solutions y(t) also.

Step 1: General form

Mass–spring oscillator equation: 

 $$\begin{gathered} {\mathrm{F}}_{\mathrm{ext}}=\left[\mathrm{inertia}\right]\mathrm{y}+\left[\mathrm{damping}\right]\mathrm{y}\text{'}+\left[\mathrm{stiffness}\right]\mathrm{y} \\ =\mathrm{my}+\mathrm{by}\text{'}+\mathrm{ky} \end{gathered}$$.......(1)

 

Step 2: Evaluate the given equation

Given that, $\mathrm{y}-6{\mathrm{y}}^2=0$ and $\mathrm{y}\left(\mathrm{t}\right)=\frac{1}{{\left(\mathrm{c}-\mathrm{t}\right)}^2}$

 

Compare the given equation with equation (1) to get stiffness.

 

Here, the stiffness is negative $\mathrm{k}=-6\mathrm{y}$ and negative sign indicates it lends to reinforce, rather than oppose, displacement. 

 

And the solutions of y(t) grow rapidly with the positive time.

 

Then, the stiffness constant helps in explaining the runaway behaviour of the solutions also.