# Differential Equation - SOLDE

1. Mar 5, 2007

### DieCommie

First let me say Im very frustrated at this and could use some real through advice. My school says to not take differential equations for a BS in physics. They instead give us a real poor packet to learn it on our own with 'guidance' from a teacher. :uhh: I am wishing I took diff. eq. anyway...

1. The problem statement, all variables and given/known data
$$y\ddot - y\dot -20 y = 17sin(3t)$$

1) Find the general soltuon for homogenous
2)find particular soltuons
3)fin solution for $$y\dot(0) = -2$$, $$y(0) = -1$$
2. Relevant equations

3. The attempt at a solution
First I need to solve the homo. part. I think I can do that.
I find the characteristic equation to be $$r^2 -r -20 = 0$$ and get $$y(t) = \alpha e^(2t) + \beta e^(-t)$$

I now need to find a particular solution. I have no idea how to do this. A table in my 'packet' says for inhomogeneity of C sin(wt) , the general form of $$y_p(t)$$ is $$A cos (\omega t) + B sin (\omega t)$$.

What do I do with this $$y_p(t)$$ thing? Please give me some very detailed hints on this part! Thank you.

If I can get the particular solution I think I can apply the boundary conditions.

Last edited: Mar 5, 2007
2. Mar 5, 2007

### Dr Transport

Check your homogeneous solutionagain, you may have typed in a mistake. As for the particular solution, $$\omega = 3$$ so plug in your solution and match coefficients to obtain a final soluiton.

3. Mar 5, 2007

### DieCommie

Ok I found small mistake and now have the correct homo. soltution. $$y(t) = \alpha e^(5t) + \beta e^(-4t)$$.

I am still quite lost as what to do with the particular solution...

I have to add it to the homo. solution? Do I just ignore the homo. solution? I know omega = 3, you say 'so plug in your solution and math coefficents' Do set the particular soltion equal to the homo. solution? What coefficents do I match...
I this what you mean ?$$\alpha e^5t + \beta e^-4t = Acos(3t) + B sin(3t)$$

4. Mar 6, 2007

### HallsofIvy

Staff Emeritus
As long as the "right hand side" does not involve any functions that also satisfy the homogeneous equation, yes, you can "ignore" the solution to the homogeneous equation.

Find the first and second derivatives of Acos(3x)+ Bsin(3x), plug them into the equation and solve the resulting equations for A and B.

You do NOT want
$$\alpha e^5t + \beta e^-4t = Acos(3t) + B sin(3t)$$