From 35cc260200f5998c64d7a9c9076399d5885796a6 Mon Sep 17 00:00:00 2001
From: Scarlett Gauthier <s.s.gauthier@student.tudelft.nl>
Date: Thu, 6 Aug 2020 13:56:48 +0000
Subject: [PATCH] Add fifth page of lecture notes.

---
 src/differential_equations_1.md | 70 ++++++++++++++++++++++++++++++++-
 1 file changed, 69 insertions(+), 1 deletion(-)

diff --git a/src/differential_equations_1.md b/src/differential_equations_1.md
index dad856f..21545ba 100644
--- a/src/differential_equations_1.md
+++ b/src/differential_equations_1.md
@@ -312,7 +312,75 @@ only if they are linearly independent for fixed $t$:
 
 $$det \big{(}**\phi**_1 (t) | **\phi**_2 (t) | \cdots | **\phi**_n (t) \big{)} \neq 0.$$
 
-If this condition holds for one $t$, it holds for all $t$. 
+If this condition holds for one $t$, it holds for all $t$.
+
+## Inhomogeneous systems ##
+
+In addition to the homogeneous equation, an inhomogeneous equation has an 
+additional term, which may be a funcction of the independent variable. 
+
+$$ \dot{**x**}(t) = **A**(t) **x**(t) + **b**(t).$$
+
+There is a simple connection between the general solution of an inhomogeneous 
+equation and the corresponding homogeneous equation. If $**\psi**_1$ and $**\psi**_2$
+are two solutions of the inhomogeneous equation, then their difference is a 
+solution of the homogeneous equation 
+
+$$(\dot{**\psi**_1}-\dot{$**\psi**_2$}) = **A**(t) (**\psi**_1 - **\psi**_2). $$
+
+The general solution of the inhomogeneous equation can be written in terms of 
+the basis of solutions for the homogeneous equation, plus one particular solution
+to the inhomogeneous equation,
+
+$$**x**(t) = **\psi**(t) + c_1 **\phi**_1 (t) + c_2 **\phi**_2 (t) + \cdots + c_n **\phi**_n (t). $$
+
+In the above equation, $\{**\phi**_1 (t), **\phi**_2(t), \cdots, **\phi**_n (t) \}$
+form a basis for the solution space of the homogeneous equation and $**\psi**(t)$
+is a particular solution of the inhomogeneous system. 
+
+Now we need a strategy for finding the solution of the inhomogeneous equation. 
+Begin by making an ansatz that $**x**(t)$ can be written as a linear combination 
+of the basis functions for the homogeneous system, with coefficients that are 
+functions of the independent variable. Ansatz:
+
+$$**x**(t) = c_1(t) **\phi**_1 (t)+ c_2(t) **\phi**_2(t) + \cdots + c_n(t) **\phi**_n (t) $$
+
+Define the vector $**c**(t)$ and matrix $**\Phi**(t)$ as
+
+$$**c**(t) = \begin{bmatrix}
+c_1(t) \\
+\vdots \\
+c_n(t) \\
+\end{bmatrix} $$
+$$**\Phi**(t) = \big{(} **\phi**_1 (t) | \cdots | **\phi**_n (t) \big{)} $$
+
+With these definitions, it is possible to re-write the ansatz for $**x**(t)$,
+
+$$ **x**(t) = **\Phi**(t) **c**(t).$$
+
+Using the Leibniz rule, we then have the following expanded equation,
+
+$$\dot{**x**}(t) = \dot{**\Phi**}(t) **c**(t) + **\Phi**(t) \dot{**c**}(t).$$
+
+Substituting the new expression into the differential equation gives,
+
+$$\dot{**\Phi**}(t) **c**(t) + **\Phi**(t) \dot{**c**}(t) = **A**(t) **\Phi**(t) **c**(t) + **b**(t) $$
+$$**\Phi**(t) \dot{**c**}(t) = **b**(t). $$
+
+In order to cancel terms in the previous line we made use of the fact that 
+$**\Phi**(t)$ solves the homogeneous equation $\dot{**\Phi**} = **A** **\Phi**$.
+
+By way of inverting and integrating, we can write an equation for the coefficient
+vector $**c**(t)$
+
+$$**c**(t) = \int **\Phi**^{-1}(t) **b**(t) dt.$$
+
+With access to a concrete form of the coefficient vector, we can then write down
+the particular solution,
+
+$$**\psi**(t)= **\Phi**(t) \cdot \int **\Phi**^{-1}(t) **b**(t) dt .$$
+
+
 
 
 
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