How to Solve a Non-Homogeneous Laplace Equation?

[reece]

Hi, I can solve homogeneous laplace fine, but with RHS i get stuck half way through.

Q:
y' +3y = 8e$$^{t}$$
y(0) = 2

Working as if it was homogeneous..

sY(s) - 2 + 3Y(s) = 8 . $$\frac{1}{s-1}$$
Y(s) (s+3) - 2 = 8 . $$\frac{1}{s-1}$$

I think the next step is
Y(s) = $$\frac{2}{s+3}$$ + $$\frac{8}{s-1}$$
and then do partial fractions but i don't think it leads me to where I need to be. I think i need to make it into a heaviside ??

Any help would be great. thanks

 

[coomast]

Hello reece,
It took me a few minutes to figure out what was going on. Your method is OK, except that you should rewrite the final line as:
$$(s+3)Y(s)=\frac{8}{s-1}+2=2\cdot \frac{s+3}{s-1}$$
From which:
$$Y(s)=\frac{2}{s-1}$$
And thus the final solution is:
$$y(t)=2e^t$$

I expected two exponentials and therefore it is interesting to solve it without using Laplace. The solution is then
$$y(t)=A e^{-3t}+2e^t$$
After applying the boundary condition you get A=0.

It is a strange equation because the exponential is unbounded for large t.

 

[tacman]

A: Hi there,

It looks like you're on the right track! After finding the partial fraction decomposition, you'll end up with the following equation:

Y(s) = \frac{2}{s+3} + \frac{8}{s-1} = \frac{A}{s+3} + \frac{B}{s-1}

where A and B are constants. To solve for these constants, you can use the method of undetermined coefficients. This involves setting up a system of equations using the given equation and its derivatives, and then solving for the constants.

Once you have the values for A and B, you can substitute them back into the equation and take the inverse Laplace transform to find the solution to the non-homogeneous equation. Don't worry about the Heaviside function - that is used for solving systems of equations with step functions, but it is not necessary for this problem.

Hope this helps!