Finish final case of page 9.

src/differential_equations_1.md

@@ -738,9 +738,38 @@ $$**x**(t) = c_1  e^{\lambda_1 t} **v**_1 + c_2(t e^{\lambda_1 t} **v**_1 + e^{\
     1 \\
     \end{bmatrix}\big{)}.$$
 
+### Case 3: Higher multiplicity eigenvalues###
+In this case we consider the situation where the matrix $**A**$ has an 
+eigenvalue $\lambda$ with multiplicity $m>2$, and only one eigenvector $**v**$ 
+corresponding to $\lambda$, $(**A** - \lambda \mathbbm{1})**v**=0$. In this case notice 
+that $**A**$ must be at least an $m \times m$ matrix. 
 
+To solve such a situation, we will expand upon the result of the previous 
+section and define the vectors $**v**_2$ through $**v**_{m}$ by
 
-$\begin{bmatrix} \end{bmatrix}$
+$$(**A**- \lambda \mathbbm{1}) **v**_2 = **v**_1$$
+$$\vdots$$
+$$(**A**- \lambda \mathbbm{1}) **v**_m = **v**_{m-1}.$$
+
+Then, the subset of the basis of solutions corresponding to eigenvalue $\lambda$
+is formed by the vectors
+
+$$**\phi**_{k}(t) = e^{\lambda t} \big{(} \frac{t^{k-1}}{(k-1)!}**v**_1 + \cdots + t **v**_{k-1} + **v_{k}** \big{)} \ \forall k \epsilon \{1, \cdots, m \}.$$
+
+To prove this, first take the derivative of $**\phi**_{k}(t)$,
+
+$$\dot{**\phi**_{k}(t)} = \lambda **\phi**_{k}(t) + e^{\lambda t} \big{(} \frac{t^{k-2}}{(k-2)!}**v**_1 + \cdots + **v**_{k-1} \big{)}.$$
+
+Then, for comparison, mulitply $**\phi**_k(t)$ by $**A**$
+
+$$**A** **\phi**_k (t) = e^{\lambda t} \big{(} \frac{t^{k-1}}{(k-1)!}\lambda **v**_1 + \frac{t^{k-2}}{(k-2)!} **A** **v**_2 + \cdots + **A** **v**_{k-1} + **A** **v**_k \big{)}$$
+$$**A** **\phi**_k (t) = \lambda **\phi**_k (t) + e^{\lambda t} \big{(} \frac{t^{k-2}}{(k-2)!}(**A**- \lambda \mathbbm{1})**v**_2 + \cdots + t (**A**- \lambda \mathbbm{1})**v**_{k-1} + (**A**- \lambda \mathbbm{1})**v**_{k}  \big{)}$$
+$$**A** **\phi**_k (t) =  \lambda **\phi**_k (t) + e^{\lambda t} \big{(} \frac{t^{k-2}{(k-2)!} **v**_1 + \cdots + t **v**_{k-2} + **v**_{k-1} \big{)}$$
+$$**A** **\phi**_k (t) = \dot{**\phi**}_{k}(t).$$
+
+Notice that in the second last line we made use of the relations 
+$(**A**- \lambda \mathbbm{1})**v**_{i} = **v**_{i-1}$. This completes the proof 
+since we have demonstrated that $**\phi**_{k}(t)$ is a solution of the DE.