Use complex valued functions for

clearvars

We want to solve the ODE

For the homogeneous problem we obtain the characteristic polynomial

syms p(r) % define symbolic variable r and function p(r)

p(r) = r^2+2*r+1

p(r) =

Solving

gives

solve(p(r)==0)

ans =

, hence

For the particular solution we use the method of undetermined coefficients:

Plugging this into the ODE gives two equations for

. We solve this linear system and obtain our particular solution.

Alternative method using complex

We use one complex number

and

We have

We consider the complex-valued solution

of the ODE

and find a particular solution

. Then

will be a particular solution for

Plugging

into

gives

hence

C = (2-1i)/p(2i)

C =

yielding

Here is the solution of the ODE found by dsolve:

syms y(t) % define symbolic variable t and function y(t)

Dy = diff(y); D2y = diff(Dy);

sol = dsolve(D2y+2*Dy+y==2*cos(2*t)+sin(2*t));

simplify( sol , 1000)

ans =