# Thread: help with differential equations

1. ## help with differential equations

Hi there , i have the following problam , and i sort of got stuck in the middle
anyways i have a system of 2 differential equations so here it is

y'=y-10z
z'=y-z+3cos(2t)

seems fairly simple , anyways i reached a point where i have Z(s) and Y(s) via laplace transfrom and well , got a bit stuck with it. so to save you time

Y(s) = (s^3 + S^2 -26s +4)/((s^2 + 4)(s^2 + 9))
Z(s) = (2s^2 - 3s +4)/((s^2 +4)(s^2 + 9))

i am fairly sure of the Y(s) and Z(s) i've recieved so anyone kind enough to help , might as well skip the middle part of reaching those two. anyways my request is a way to simplify Y(s) and Z(s) to the point where i can do L^-1[Y(s)] and L^-1[Z(s)] and ofc the way it was done. help would be much appriciated.

2. Partial fractions right?

$\frac{g(s)}{(s^2+4)(s^2+9)}=\frac{as+b}{s^2+4}+\fr ac{cs+d}{s^2+9}$

Code:
In[13]:=
LaplaceTransform[InverseLaplaceTransform[
(s^3 + s^2 - 26*s + 4)/((s^2 + 4)*
(s^2 + 9)), s, t], t, s]

Out[13]=
(1/3)*(-((18*s)/(4 + s^2)) + 3/(9 + s^2) +
(21*s)/(9 + s^2))
. . . what, I ain't proud.

3. Thanks Shawsend , been a while since i've done any partial fractions , wasnt sure if g(s) could be just about any function or if it was restricted somehow.

4. Degree of denominator > degree of numerator for partial fractions. Also, just learned could use $\text{Apart}$ command in Mathematica for this:

Code:
In[17]:=
Apart[(s^3 + s^2 - 26*s + 4)/
((s^2 + 4)*(s^2 + 9))]

Out[17]=
-((6*s)/(4 + s^2)) + (1 + 7*s)/(9 + s^2)
Caveat: I do not wish to promote using Mathematica until you're comfortable with doing it yourself first.