Where and where f and g are linear functions of x and y. The solution is found by differentiating one of the equations and then substituting to remove one of the variables and leave a second-order linear differential equation. For example, if and (B) with x=1, y=4 when t=0. Then writing (A) as and differentiating gives
So substituting C and D into B gives ⇒
The solution to this is x=Ae−3t+Be2t, y=−4Ae−3t+Be2t and substituting for the boundary conditions gives A=1 and B=0 and the solutions to the original differential equations are x=e−3t, y=−4e−3t.