This is the last part of an example I did in class. The problem was y'' +3y' + 2y = g(t) where g(t) was 0 until t=10; from 10 to 20 it was linear, t-10; after 20 it was zero again. In class we found an expression for L(y). Here I use a computer to find y = inverse Laplace transform of L(y), and then I plotted it. (I used Maple since MATLAB does not have an easy way to export to HTML, but the MATLAB commands are similar.)

> with(inttrans):

> invlaplace((exp(-10*s)*(1/s^2) - exp(-20*s)*(1/s^2 + 10/s))/(s^2+3*s+2),s,t);

> f:= t -> (1/2*t-23/4-1/4*exp(-2*t+20)+exp(-t+10))*Heaviside(t-10) + (-1/2*t+23/4-19/4*exp(-2*t+40)+9*exp(-t+20))*Heaviside(t-20);

> plot(f(t),t=0..30,thickness=3,color=blue);

Compare the shape of the solution y(t) with the shape of g(t). What would happen if g(t) from 10 to 20 was periodic, say sin(t) ? What if g(t) was a constant or t^2 from 10 to 20? In this case the homogeneous solution goes to zero. What would y(t) look like if the homogeneous solution went to infinity? What if the homogeneous solution was periodic? Construct such examples and see if your conjectures are correct. The best way to learn any subject is to play with it!