ENGI 3424 - Engineering Mathematics
Laplace Transforms

Below are the commands from the Maple worksheet.
Open the worksheet in Maple to see the resulting expressions.

>	with(inttrans):
	(needed for Maple to understand many of the following commands)

 

Example 3.05.1 [part:  inverse of   Y ( s )   to find the complete solution   y ( t)   of an initial value problem] 

>	F(s,w) := (s-5)/((s-2)*(s-3));

 

(1)

 

>	invlaplace(F(s,w), s, t);

 

(2)

 

Example 3.05.2 [part:  inverse of   Y ( s )  to find the complete solution   y ( t )   of an initial value problem] 

>	F(s,w) := (5*s^2+11*s-10)/((s+4)*(s^2+4*s+13));

 

(3)

 

>	invlaplace(F(s,w), s, t);

 

(4)

 

Example 3.06.1 - a standard inverse 

>	F(s,w) := 1/(s*(s^2 + w^2));

 

(5)

 

>	invlaplace(F(s,w), s, t);

 

(6)

 

Example 3.06.2 - a standard inverse 

>	F(s,w) := 1/(s^2*(s^2 + w^2));

 

(7)

 

>	invlaplace(F(s,w), s, t);

 

(8)

 

Example 3.07.2 - Laplace transform involving the Heaviside unit step function 

>	f(t) := Heaviside(t-4)*(t-4)^3;

 

(9)

 

>	laplace(f(t), t, s);

 

(10)

 

Example 3.07.3 - inverse Laplace transform involving the Heaviside unit step function 

>	F(s,w) := exp(-5*s)/(s^2 + 4);

 

(11)

 

>	invlaplace(F(s,w), s, t);

 

(12)

 

Example 3.07.5 - inverse Laplace transform involving the Heaviside unit step function 

>	F(s,w) := exp(-3*s)*(3*s+1)/(s^2*(s^2 + 4));

 

(13)

 

>	invlaplace(F(s,w), s, t);

 

(14)

 

[which is equivalent to (3/4 + ( t -3)/4 - 3/4 cos2( t -3) - 1/8 sin2( t -3)) H ( t -3) ] 

Example 3.10.2 - a standard Laplace transform 

>	f(t,w) := t*sin(w*t);

 

(15)

 

>	laplace(f(t,w), t, s);

 

(16)

 

Example 3.11.2 - a standard inverse Laplace transform 

>	F(s,w) := 1/(s^2 + w^2)^2;

 

(17)

 

>	invlaplace(F(s,w), s, t);

 

(18)

 

Maple Worksheet   ex3.mws