Proof of general solution or method to find the general solution of higher order differential equations

Solve the following differential equation:

$$x^{(n)}+a_1x^{(n-1)}+...+a_{n-1}x'+a_nx=b(t)$$

I know that first I have to find the solution of the homogenous version of this differential equation and then the solution would be $$x(t) = x_o(t)+x_p(t)$$

where $$x_o(t)$$ is the solution of the homogenous DE and $$x_p(t)$$ is a particular solution that you have to 'observe', right? What is that and what is the intuition behind the $$x_p$$? What I mean is, I need more details about how to solve this kind of DE.

• Any textbook on differential equations should have a rather extensive section on linear DE and linear systems of first order DE. What text did you use and how far did you get? Do you understand how a simple linear system $Ax=b$ has an affine solution space that is a parallel copy of the homogeneous solution space or kernel of $A$? – LutzL Jan 10 at 9:36
• @LutzL um.. no I don't think I understood that but when I saw and got over first order ode I thought we should first get to higher order rather than linear systems? – C. Cristi Jan 11 at 15:59

Think about the discretized equation for a time discretization $$t_k=t_0+kh$$, $$t_{final}=t_N$$, $$x_k\approx x(t_k)$$, replacing derivatives with difference quotients of the same order.
Then the resulting difference equation is an equation for $$x_{k-n},...,x_{k-1}, x_k$$, for $$k=n,...,N$$. You get thus a linear system of $$N+1-n$$ linear equation for $$N+1$$ variables. The solution space is an affine manifold parallel to the solution space of the homogeneous equations of dimension $$n$$. Which means that any point of the solution space can be obtained as a fixed, particular, solution plus some homogeneous solution.
The points in that space are determined by $$n$$ parameters, these can be connected to the initial values for $$x(t_0), x'(t_0),...,x^{(n-1)}(t_0)$$.