How do I solve: p(t)f''(t)+q(t)f'(t)=Kf(t) ?

[simplex]

How do I solve the equation: p(t)f''(t)+q(t)f'(t)=Kf(t) ?

I have the equation: p(t)f''(t)+q(t)f'(t)=Kf(t) that results from a problem of physics.
- p(t) and q(t) are two known periodical functions of period T
- f(t) is an unknown function
- K is an unknown constant

What I need is to determine all the possible values of these K constants with the intention of finding the real ones if they exists.

 

[wywong]

Have you tried Fourier series like this?

$$f(t) = \sum {a_n cos(\frac{n\pi t}{T})+b_n sin(\frac{n\pi t}{T})$$

Note that the fundamental frequency of the above series is half of that of p(t) and q(t).

 

[matematikawan]

How do I solve the equation: p(t)f''(t)+q(t)f'(t)=Kf(t) ?

I have the equation: p(t)f''(t)+q(t)f'(t)=Kf(t) that results from a problem of physics.
- p(t) and q(t) are two known periodical functions of period T
- f(t) is an unknown function
- K is an unknown constant

What I need is to determine all the possible values of these K constants with the intention of finding the real ones if they exists.

If we let y(t)=f(t) then your equation is a linear homogeneous DE.
p(t)y''+q(t)y' - Ky = 0.

But you are not particularly interested with the solution f(t), aren't you ?
The existence theorem may be useful here I think! What is the initial condition?

 

[flatmaster]

I agree with wywrong in attempting a Fourier series solution. However, I would use the complex exponential form of the Sin and Cos to make your math cleaner.

http://en.wikipedia.org/wiki/Fourier_series

 

[flatmaster]

Out of curiousity, where's the physics? This is math.

 

[wywong]

I agree with wywrong in attempting a Fourier series solution. However, I would use the complex exponential form of the Sin and Cos to make your math cleaner.

Flatmaster is right. Complex exponential form is better. And I think whatever k's value is, there is a general solution of the form

$$f(t)=\sum{a_n e^{(r_n+in\omega)t}}$$

where $$\omega=\pi/T$$.

For $$r_n<0$$, those components will die down. If there are $$r_n>0$$, the system will be unstable as those components will increase exponentially. For frequency components with $$r_n=0$$, they will be the steady state solutions. To be stable, k can assume a large range of values. To have steady state solutions, k can only assume discrete values. For those k's any slight change in k or p or q may render the system unstable, unless your system is non-linear beyond certain amplitude.

P.S. My username is wywong, not wywrong B:).

 

[simplex]

Thank you. Your idea with using Fourier series was good. Finally I investigated this method (reading documentation on the internet) and I come up with continuous domains of K for which the equality p(t)f''(t)+q(t)f'(t)=Kf(t) can take place.

For instance if:
K=[0 1] -> f(t) exists and it is not zero
K=(1 4] -> f(t) does not exists
K=(4-8] -> f(t) exists and it is not zero