Given:
where 
With initial conditions: 
Find the overall solution 
for 
and plot these solutions on the interval from ![{\displaystyle [-{\frac {3}{4}},3]\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/308e17971be2299355ca353c40d0f3ee8007b72a)
First we find the homogeneous solution to the ODE:
The characteristic equation is:
Then,
Therefore the homogeneous solution is:
Now to find the particulate solution
For n=4
We can then use a matrix to organize the known coefficients:
Then, using MATLAB and the backlash operator we can solve for these unknowns:
Therefore
Superposing the homogeneous and particulate solution we get
Differentiating:
Evaluating at the initial conditions:
We obtain:
Finally we have:
For n=7
We can then use a matrix to organize the known coefficients:
Then, using MATLAB and the backlash operator we can solve for these unknowns:
Therefore
Superposing the homogeneous and particulate solution we get
Differentiating:
Evaluating at the initial conditions:
We obtain:
