Please help me solve the following with steps:
Solve the differential equation by method of Laplace transform. (t) + y(t) = et (t) + y (t) = t (t) - 0.2 (t) + y(t) =0 y(0) = 1; (0) = 2 y(0) = 1; (0) = -1 y(0) = 1; (0) = 1
Solution
L(y\'\') = s^2*Y(s) - s*y(0) - y\'(0)
L(y\') = s*Y(s)-y(0)
L(e^t) = 1/(s-1)
L(t) = 1/s^2
a)
s^2*Y(s) - s*y(0) - y\'(0) + Y(s) = 1/(s-1)
(1+s^2)*Y(s) - s - 2 = 1/(s-1)
Y(s) = {(s+2)*(s-1)+1}/(1+s^2)
b)
s^2*Y(s) - s*y(0) - y\'(0) + Y(s) = 1/s^2
(1+s^2)*Y(s) - s + 1 = 1/s^2
Y(s) = {s^2*(s-1)+1}/(s^2*(1+s^2))
c)
s^2*Y(s) - s*y(0) - y\'(0) -0.2*( s*Y(s) - y(0)) + Y(s) = 0
s^2*Y(s) - s - 1 - 0.2*s*Y(s) + 0.2 + Y(s)
Y(s) = (s+0.8)/(s^2 - 0.2*s + 1)
.
