## Solving an Initial Value Second Order Non Homogeneous Linear Differential Equation With Constant Coefficients

Any constant coefficient non homogeneous linear differential equation with initial or boundary conditions can be solved by finding the complementary solution
$y_c$
of the homogeneous equation and any solution
$y_p$
of the non homogeneous equation, then the general solution is
$y=y_c+y_p$
. The solution that then be fitted to the initial or boundary conditions to find the values of the arbitrary constants.
Example: Solve the equation
$\ddot{y}+4y=3$
with initial conditions
$y(\pi)=1, \: \dot{y}(\pi)=1$
.
The homogeneous equation is
$\ddot{y}+4y=0$
with solution
$y_c=Acos 2t+Bsin 2t$
.
The non homogeneous equation
$\ddot{y}+4y=3$
has a solution (found by putting
$y=C$
)
$y=\frac{3}{4}$
.
The general solution is then
$y=Acos 2t+Bsin2t + \frac{3}{4}$
.
$y(\pi)=1 \rightarrow A+ \frac{3}{4} =1 \rightarrow A= 1-\frac{3}{4}= \frac{1}{4}$

$\dot{y}(\pi)=2 \rightarrow 2B =2 \rightarrow B= 1$

Then
$y=\frac{1}{4}cos2t+sin 2t+ \frac{3}{4}$
.