(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Hi this is my first post here... just started a new course, and in the prep material I've been given this question which is doing my head in!

The DE

[tex]\frac{d^2 x}{dy^2}+\frac{dx}{dt}+x=sin(wt)[/tex]

describes the deflection x of some suspension as a function of t, in response to a bumby surface. Here w>0 is a parameter that describes the frequency of the bumps.

(hopefully the LaTeX equation looks ok, because for some reason this works computer doesn't want to show me it!)

2. Relevant equations

The first question asks me to solve the Associated Homogeneous equation,

for which I got [tex]-\frac{1}{2}+-\sqrt{\frac{3}{4}}i[/tex] as a conjugate pair of solutions of the aux. equation. This gives me the General solution,

[tex]x_{c}=e^{-1/2t}(C cos\sqrt{\frac{3}{4}}t+D sin\sqrt{\frac{3}{4}}t)[/tex]

Hopefully I'm all good so far?

3. The attempt at a solution

So the next part asks me to find a particular integral which satisfies the DE and hence obtain the general solution...(I should add, I've only ever learn't the method of undertermined coefficients from previous courses, so the method can't be to exotic.)

So I guessed a solution of the form:

[tex]x=p cos(wt)+q sin(wt)[/tex]

Differentiating (twice), we end up with

[tex]\frac{dx}{dt}=-wp sin(wt)+wq cos(wt)[/tex]

and

[tex]\frac{d^2 x}{dt^2}=-w^2 p cos(wt)-w^2 q sin(wt)[/tex]

I then substituted these back into the equation, which after some jiggery pokery I got

[tex](-w^2 p+wq+p)cos(wt)+(-w^2 q-wp+q)sin(wt)=sin(wt)[/tex]

So

[tex](-w^2 p+wq+p)=0[/tex]

and

[tex](-w^2 q-wp+q)=1[/tex]

But now I'm stuck, how do I find the values for p and q? Can anyone give me a clue as to how to find them?

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# Homework Help: Particular integral problem for a Differential Equation.

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