Exponentially driven harmonic oscillator

[vbrasic]

Homework Statement

An un-damped harmonic oscillator natural frequency $\omega_0$ is subjected to a driving force, $$F(t)=ame^{-bt}.$$ At time, $t=0$, $x=\dot{x}=0$. Find the equation of motion.

Homework Equations

$F=m\ddot{x}$

The Attempt at a Solution

We have $$m\ddot{x}+kx=F(t)=ame^{-bt}.$$ Dividing through by $m$ we have $$\ddot{x}+\omega_0^2x=ae^{-bt}.$$ From a course in differential equations, I know that the solution to this is the sum of the homogenous solution with the particular solution. The homogenous solution is of the form $$A\cos{\omega_0t}+B\sin{\omega_0t}.$$ I am having a bit of trouble finding the particular solution. Because the solution is an exponential, I assume a solution of the form $x(t)=ce^{-bt}.$ Differentiating twice, we have $$\ddot{x}=b^2ce^{-bt}.$$ Plugging into the expression gives, $$b^2ce^{-bt}+\omega_0^2ce^{-bt}=ae^{-bt}.$$ From this we have that $$a=b^2c+\omega_0^2c.$$ I'm not sure if I'm on the right track here. Any help would be appreciated.

 

[Orodruin]

First of all, you have a typo in your differential equation. I suggest you check the dimensions of each term to find it.

Plugging into the expression gives, $$b^2ce^{-bt}+\omega_0ce^{-bt}=ae^{-bt}.$$ From this we have that $$a=b^2c+\omega_0c.$$ I'm not sure if I'm on the right track here. Any help would be appreciated.

What is your unknown in finding the particular solution? Can you find it from your equation? Does what you get solve the differential equation?

 

[vbrasic]

First of all, you have a typo in your differential equation. I suggest you check the dimensions of each term to find it.

Should have been $\omega_0^2.$ Also, I'm not solving for $a$. Rather I should be solving for the $c$ of the particular solution. So from the $$(b^2+\omega_0^2)c=a,$$ we get $$c=\frac{a}{b^2+\omega_0^2}.$$ We verify this is indeed the particular solution. So the most general solution is, $$A\cos{\omega_0t}+B\sin{\omega_0t}+\frac{a}{b^2+\omega_0^2}e^{-bt}.$$ From here we would apply the initial conditions?

 

[Orodruin]

Indeed, applying the initial conditions will fix your unknown constants $A$ and $B$.