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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.

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  • $\begingroup$ 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 $A$? $\endgroup$ – LutzL Jan 10 at 9:36
  • $\begingroup$ @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? $\endgroup$ – C. Cristi Jan 11 at 15:59
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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 $x(t_0), x'(t_0),...,x^{(n-1)}(t_0)$.

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