Solving ODE with Laplace, with product of two functions

Question

How would I attack the following problem in order to solve it with Laplace transforms?

$$y'''-y=-ye^{2t},\quad y(0)=0,\quad y'(0)=-1,\quad y''(0)=-3.$$

My issue is that I'm not sure how to apply the Laplace transform when there is a product of two functions.

Answer 1

Well, there's good news, and there's bad news. The good news is that $$L\left[-ye^{2t}\right]=-Y(s-2).$$ But that's also the bad news, because now your DE becomes $$s^3 Y(s)+s+3-Y(s)=-Y(s-2).$$ This is now a recurrence relation with no particularly easy solution in sight. If you just run the following in Mathematica:

DSolve[{y'''[t]-y[t]==-y[t] Exp[2t],y[0]==0,y'[0]==-1,y''[0]==-3},y[t],t]

you get an immensely complicated expression involving HypergeometricPFQ. You can simplify it a bit if you run //N at the end of the DSolve command, but that produces an approximation. The result Mathematica yields has real and imaginary parts.

So you can solve this in terms of the HypergeometricPFQ function and the exponential; it's extremely complicated.