Example for forcing function with Dirac deltas
Solve the initial value problem
clearvars
syms s t Y
f = 2*dirac(t-1)-3*dirac(t-2)
Matlab plots dirac(t) as a zero function, so we don't need to plot the function f.
F = laplace(f)
Y1=s*Y-0; Y2 = s*Y1-0;
Sol = solve(Y2+3*Y1+2*Y==F,Y)
partfrac(1/(s^2+3*s+2))
sol = ilaplace(Sol)
fplot(sol,[0 6]); xlabel('t'); title('solution y(t)'); ax_origin
We can see that at 
the function 
has a jump of 2, and at 
the function has a jump of 
.
the function has a jump of 2, and at the function has a jump of .fplot(diff(sol),[0 6]); xlabel('t'); title('derivative y''(t)'); ax_origin
simplify( diff(sol) )
Draw the x and y axes through the origin:
function ax_origin
set(gca,'XAxisLoc','origin','YAxisLoc','origin'); box off
end