Understanding Engineering Mathematics

(i)

2 s^2 − 3 s+ 1

2 s^3

(ii)

1

s+ 1

(iii)

1

s− 3

(iv)

4

2 s− 1

(v)

5

s^2 + 4

(vi)

3 s

s^2 + 9

(vii)

s+ 1

s^2 − 5 s+ 6

(viii)

1

(s+ 1 )(s+ 2 )(s+ 3 )

Answer

(i) 1−^32 t+^14 t^2 (ii) e−t (iii) e^3 t (iv) 2et/^2

(v)^52 sin 2t (vi) 3 cos 3t (vii) 4e^3 t− 3 e^2 t

(vi)^12 e−t−e−^2 t+^12 e−^3 t

17.6 Solution of initial value problems by Laplace transform

The inverse Laplace transform enables us to solve linear initial value problems. With what
you now know about the inverse transform, try this on your own.

Problem 17.8

Solve the initial value problem

y′Yy= 1 y. 0 /= 1

by Laplace transform.

Taking the Laplace transform of the equation:

L[y′+y]=L[y′]+L[y]

=sy(s) ̃ −y( 0 )+ ̃y(s)

=(s+ 1 )y(s) ̃ − 1 =L[1]=

1

s

Hence

y(s) ̃ =

1

s

so the solution is

y(t)= 1

which you can check by direct solution of the equation.
The above example illustrates the general approach to solving linear initial value prob-
lems by Laplace transform: