Solve yyex where y00y00y00 using laplace transformsSolution

Solve y\'\'+y\'=e^x , where y(0)=0,y\'(0)=0,y\'\'(0)=0 using laplace transforms

Solution

y\'\'+y\' = e^x

The laplace transform of y

Y = Y(s) = L{Y(s)} S

Then

S^2 Y - 1 + SY = 1 / (S-1)

Hence Y = S / (S-1)(S^2 + S )

Y = S / (S-1) S (S+1)

Y = A / (S-1) + B / S + C / (S+ 1 )

A ( S(S+1)) + B ((S_1)(S+1) + C ( S(S-1)) = S

substituting S = -1

C ( -1 (-2)) = S

2 C = -1

C = -1/2

substitute S = 1

2A = 1

A= 1/2

substituting S= 0

B = 0

Thus Y = 1/2(S-1) - 1/2(S+1)

on applying laplace transform

Y = 1/2 (e^x) - 1/2(e^-x)

= e^x / 2 - e^-x /2