Solving for an unknown in differential equation solution

[Ewan_C]

Homework Statement

(1): $$\frac{d^{2}y}{dx^{2}} + 4\frac{dy}{dx} + 20y = 10e^{-2x}\cos (rx)$$, $$y(0) = 1$$, $$y'(0) = 0$$

Show that there is precisely one value for r for which $$e^{2x}y(x)$$ can become arbitrarily large where $$y(x)$$ is a solution of (1)

The Attempt at a Solution

I used Maple to solve the DE and found a particular solution $$y(x)$$.

$$e^{2x}y(x) = -\frac{10(-\cos 4x + \cos rx)}{-16+r^2}$$

When r tends to 4 or -4 then $$e^{2x}y(x)$$ can become arbitrarily large, but this doesn't seem to me like it satisfies the question - it's not "precisely one value" and it's not an exact value of r. I graphed $$e^{2x}y(x)$$ and can't see anywhere else it can become arbitrarily large. I feel like I'm missing something, can anyone point me in the right direction? Cheers.

 

[tiny-tim]

Hi Ewan_C!

Sorry, but this is what happens when you take short-cuts …

there is a solution for r = ±4, but it doesn't involve cos4x.

Start again, put y = ze^-2x, get the differential equation for z, put r = ±4, and look for another particular solution.

 

[Ewan_C]

Hi Tim, thanks for the response.

So you mean that I should substitute y(x) = z(x)*e^-2x into the original DE and solve it for z? Now I get it, that's what they meant from the start - I took "where y(x) is a solution of (1)" to mean a particular solution rather than the general solution.

We haven't yet been taught how to solve DEs with non-constant coefficients, so I used Maple again to solve the DE. I used the following commands:

DE := diff(exp(-2*x)*y(x),x,x)+4*diff(exp(-2*x)*y(x),x)+20*exp(-2*x)*y(x) =10*exp(-2*x)*cos(r*x);

DE_hom := diff(exp(-2*x)*y(x),x,x)+4*diff(exp(-2*x)*y(x),x)+20*exp(-2*x)*y(x) = 0;

dsolve({DE,y(0)=1,D(y)(0)=0},y(x)) - dsolve({DE_hom,y(0)=1,D(y)(0)=0},y(x));

I didn't substitute y(x)=z(x)e^-2x as it made Maple get angry with me - my Maple skills aren't too brilliant! I think what I did is equivalent though?
Am I correct in thinking that if you take the general solution of the DE and subtract the general solution of the corresponding homogeneous DE, the result is a particular solution of the DE?

This still gave me a particular solution with a cos(4x) term. I don't really know what's going on with this DE as I can't solve it by hand. Maybe the best next step would be to spend some time on the internet learning how solve DEs with non-constant coefficients?

 

[tiny-tim]

We haven't yet been taught how to solve DEs with non-constant coefficients, so I used Maple again to solve the DE. I used the following commands:

Sorry, Ewan_C, I've no idea what Maple is or how it works.

Surely you can work out what dy/dx and d²y/dx² are in terms of dz/dx and d²z/dx² without a calculator?