Two Questions: Laplace transform, general sol'n

[SurferZ]

1. I'm having trouble with the laplace transform of y"(t) + 2y'(t) + y(t) = 3te^-t, y'(0) = 4, y"(0) = 2

I worked through to Y(s) = (4s^3 + 18s^2 + 24s + 13)/(s+1)^4

i need help finding the inverse laplace transform

2. I've tried multiple times to find the general solution of the following equation always ending up at dead ends.

[x^(n+1)y^n + ay]dx + [y^(n+1)x^n + ax]dy = 0
Caution: the answer depends on the numerical value of n

Any help will be much appreciated

Thank you

 

[tkhunny]

Have you tried a partial fraction decomposition?

\(\displaystyle \L\,\frac{3}{(s+1)^{4}}\,+\,\frac{6}{(s+1)^{2}}\,+\,\frac{4}{(s+1)}\)

 

[SurferZ]

Thanks, anyone have any idea for the second question

 

[royhaas]

Try \(\displaystyle (xy)^{n+1}/(n+1) + axy = C\).

 

[SurferZ]

How does setting that at C help me? What exactly do i do with that?

 

[royhaas]

Take the derivative.