Page 80 - 4811

Solving Systems of Differential Equation Using Laplace Transform
.

and the Laplace transform would be

s 1 s
Y (s) = C 1 + C 2 + A .
2
2
2
s + 1 s + 1 s + 4
Now, let’s take the Laplace transform on both sides:
s
2
s Y (s) − 2s − 1 + Y (s) = ⇒
2
s + 4
3
2
s 2s + s + 9s + 4
2
(s + 1)Y (s) = + 2s + 1 =
2
2
s + 4 s + 4
2
3
2s + s + 9s + 4 As + B Cs + D
Y (s) = = + .
2
2
2
2
(s + 1)(s + 4) s + 1 s + 4
Comparing numerators, we get
3
2
2
2
2s + s + 9s + 4 = (As + B)(s + 4) + (Cs + D)(s + 1)
One may expand the right side and compare terms to find A, B, C and D,
but that takes more work. Let’s try by setting s into complex numbers.
3
2
Set s = i, and remember the facts i = −1, i = −i, we have
7
−2i−1+9i+4 = (Ai+B)(−1+4), 3+7i = 3B +3Ai ⇒ A = , B = 1.
3
Set now s = 2i :

1
−16i−4+18i+4 = (2Ci+D)(−3), 2i = −3D−6Ci ⇒ C = − , D = 0.
3
So
7 s 1 1 1
Y (s) = + − ,
2
2
2
3 s + 1 s + 1 3 s + 4
and
7 1
y(t) = cos t + sin t − cos 2t.
3 3
. . . .

Solving Systems of Differential Equation Using

Laplace Transform

Systems of Differential Equations. Systems of differential equations can also be
readily handled by the Laplace transform method.

Laplace transforms is also useful in solving systems of ordinary differential
equations (ODE). As we will see, the use of Laplace transforms reduces the
problem of solving a system to a problem in algebra. Let us illustrate it with a

79