Solve the given de
dx/dt=3x+2y+1
dy/dt=-2x-y+1

We can solve for first by rewriting the system of first-order ODEs as a single second-order ODE in :

Taking the derivative of the first ODE gives

while solving for gives

Then

which is linear with constant coefficients, so it's trivial to solve; the corresponding homogeneous ODE

has characteristic equation

with root (multiplicity 2), so the characteristic solution is

For the non-homogeneous ODE, assume a particular solution of the form

Substituting these into the ODE gives

Then the general solution for is

From here, we find

so that