Homogeneous Second Order ODE

#math

Homogeneous 2nd order linear equation with constant coefficients $$ay'' + by' + cy = 0 \tag{1}$$

In this case, we check the solutions of the quadratic equation $$am^2 + bm + c = 0 \tag{2}$$(called characteristic equation of $(1)$ )

Theorem

If the solutions of $(2)$ are real and distinct, say $m_{1}$ , $m_{2}$ , then $e^{m_{1} x}$ and $e^{m_{2} x}$ are the LI solutions of $(1)$ , thus the general solution is $$C_{1}e^{m_{1}x} + C_{2}e^{m_{2}x}$$
If the solutions of $(2)$ are real and equal, say $m_{1} = m_{2} = m$ , then the two LI solutions are $e^{m x}$ and $x e^{m x}$ . Thus the general solution is $$C_{1}e^{mx} + C_{2}xe^{mx}$$
If the roots are complex conjugate, $α + i β$ and $α - i β$ , then two LI solutions are $e^{α x} \sin (β x)$ and $e^{α x} \cos (β x)$ , thus the general solution is $$e^{\alpha x}(C_{1}\cos(\beta x) + C_{2}\sin(\beta x))$$

Proofs

Since we notice the constant coefficients, we guess that solution must be of the form $e^{m x}$ .

Case 1

We can clearly see that if we substitute $e^{m x}$ into $(1)$ , then we get $$(am^2 + bm + c)e^{mx} = 0$$
So clearly, if $p (m) = a m^{2} + b m + c = 0$ , then the equation is satisfied.
This means, if we have two distinct solutions $m_{1}, m_{2}$ , then $e^{m_{1} x}$ and $e^{m_{2} x}$ are solutions

Case 2

if $m_{1} = m_{2} = m$ , then $p (m) = 0$ and $p^{'} (m) = 0$ are the two conditions that help us isolate solutions.
Thus $e^{m x}$ and $$\frac{\partial}{\partial m} e^{mx} = xe^{mx}$$ are the LI solutions

Case 3

We obtain $m_{1, 2} = z, \bar{z}$ , and let $$ \begin{align} z = \alpha + i\beta \ \ y_{}(x) = e^{zx} \ \end{align} $$
Expanding, $$\begin{align} y(x) &= e^\alpha e^{i\beta x} \ \implies y(x) &= e^\alpha(\cos(\beta x) + i\sin(\beta x))\end{align}$$
We notice that this solution itself is composed of two smaller solutions, $Re (y)$ and $Im (y)$

We can see that these two must also, independently satisfy the Linear differential equation we have above, and thus, we have found one basis, thus the LI solutions are $$e^\alpha \cos(\beta x),\ \ \ \ e^\alpha \sin(\beta x)$$