Spring-Mass-Damper by Recurrence Relations

[tanky322]

Homework Statement

Solve the Mass-Spring-Damper Differential equation

mx''+bx'+kx=exp(-t)cos(t) (Where x'' is d2x/dt2 etc, don't know how to do the dots above )




I understand how to solve this problem, but the thing that confuses me is that the right is in terms of "t" and left is in terms of "x". I understand that x is a function of t such that x''=d2x/dt2, but I am confused about how to solve the equation by recurrence relations. Once I expand out each side by its series solutions and group like terms, how do I compare terms of x to terms of t?

For example, if this works out to be:

(a1+3a3)+(a2+4a2)x+(a3+6a5)x^2+...=1+t+t^2/2!+...

Can I say that a1+3a3=1; a2+4a2=1; and a3+6a5=1/2!?


Thanks!

 

[Office_Shredder]

There are no x² terms on the left, and it certainly doesn't make any sense to expand x as a power series in x, because that's just x.

The x's are functions of t. x=x(t). So to get a power series on the left, you would write out x as a power series of t, and you want to find the coefficients of the power series by comparing them to the coefficients on the right.

 

[tanky322]

This slapped me in the face about 10 minutes after I posted, made me feel rather dumb...

Thanks for your reply!