5.1. Laplace Tranform

Laplace transforms in Maple is really straightforward and doesn’t require any complicated loops like the numerical methods.

For example, let’s take the equation t^2+sin(t)=y(t) as our equation. The syntax for finding the laplace transform of this equation requires the simple syntax below:

> with(inttrans):
> laplace (t^2+sin(t)=y(t),t,s) ;

Yes, that is all. Easy, right? The syntax for the command is (equation, dependent value, independent value).

Laplace transforms and Inverse Laplace Transforms

Laplace transforms in Maple is really straightforward and doesn’t require any complicated loops like the numerical methods.

For example, let’s take the equation t^2+sin(t)=y(t) as our equation. The syntax for finding the laplace transform of this equation requires the simple syntax below:

> with(inttrans):
> laplace (t^2+sin(t)=y(t),t,s) ;

Yes, that is all. Easy, right? The syntax for the command is (equation, dependent value, independent value).

Finding the inverse Laplace transform requires a similar syntax. Just take the result we have from the Laplace transform above and apply it here:

> with(inttrans):
> invlaplace ((2/s^3)+(1/s^2+1),s,t);

Very easy indeed. The syntax for the inverse Laplace transform is reversed compared to the Laplace transform for some reason. The syntax is (equation, independent value, dependent value).

Solving equations with periodic piecewise continuous functions

> with(plots):
> with(inttrans):
> E:=1; a:=2;
> yh :=t->invlaplace((x+5+2)/(x^2+5*x+6), x, t);
> p := x^2+5*x+6;
> yp1 := t->invlaplace(E/(a*x^2*(x^2+5x+6)),x,t)
> yp2 := t->invlaplace(E*exp(-a*x)/(x*p*(1-exp(-a*x))), x, t)
> y := t-> yp1(t)-piecewise(a < t, yp2(t-a), 2*a < t, yp2(t-2*a), 3*a < t, yp2(t-3*a))
> plot(y(t), t = 0 .. 4*a);
> y2 := t->y(t)+yh(t)
> plot(y2(t), t = 0 .. 4*a);