Laplace transform for ODE with piecewise forcing function

clearvars
syms s t Y  

We want to solve the initial value problem

where the forcing function is defined as a piecewise function:

Step 1: Rewrite f(t) using step function

We can rewrite a piecewise defined function using the unit step function (Heaviside function)

as follows

Here we obtain

f = t + heaviside(t-1)*(-t);

fplot(f,[0 6]); title('forcing function f(t)')

Step 2: Find Laplace transform

We have

For the first term we get

For the second term we use the following rule:

Here

, so

Now we find

This gives

F = laplace(f)

F =

Step 3: Find Laplace transform of left hand side and solve for

We have

Y1 = s*Y-0;
Y2 = s*Y1-0;

We obtain

Sol = simplify( solve(Y2+3*Y1+2*Y==F,Y) ,1000)

Sol =

collect(Sol,'exp')

ans =

Step 4: Find inverse Laplace transform

We obtained

with rational functions

We need to find the partial fraction decomposition for

and

R1 = partfrac(1/(s^2*(s + 1)*(s + 2)))

R1 =

R2 = partfrac((-1/(s^2*(s + 2))))

R2 =

The solution

is given by

It is clear how to find

ilaplace(R1)

ans =

For

we use the rule

We first find

g = ilaplace(R2)

g =

We then obtain

In Matlab we can just apply ilaplace to Sol:

sol = ilaplace(Sol)

sol =

fplot(sol,[0 6]); title('solution y(t)'); axis auto

We can collect the terms with 'heaviside':

sol = collect(sol,'heaviside')

sol =

We can rewrite this as a "piecewise defined function":

sol = rewrite(sol,'piecewise')

sol =

We can see that the solution for

converges to zero as