I decided I wanted to revisit some techniques for solving ordinary differential equations. It's always good to through in some refreshers as I continue to write more for my blog. Here I'll focus on homogenous differential equations that have the following form:
$$ A_1 \frac{d^2 y}{dx^2} + A_2\frac{dy}{dx} - A_3y =0 .$$
The coefficients $A_1$, $A_2$, and $A_3$ are all constant. This equation can be solved using the auxiliary equation, which is derived from the fact that such differential equations have a solution of the form $y=e^{mx}$. The auxiliary equation is:
$$ f(m) = A_0 m^{n} + A_{1} m^{n-1} + \dots+ A_{n-1}m + A_{n} = 0$$
The goal is to find the roots of the auxiliary equation, and if they are real, we can use the general solution to our homogenous differential equation which is:
$$ y(x) = C_{1}e^{m_1 x}+C_{2}e^{m_2 x} + \dots + C_n e^{m_n x} $$
Let us now solve our initial differential equation at the beginning of this post, where $A_1=10$, $A_2=1$, and $A_3=-2$. We start by rewriting the equation in terms of operator power, i.e., the degree of differentiation, this would be:
$$y(10L^2+L-2)=0,$$
$L$ is the differential operator. Now we want to plug the coefficients and operator powers into the auxiliary equation above:
$$10m^{2}+m-2$$
We can find the roots for this polynomial, which are $m=-\frac{1}{2}$ and $m=\frac{2}{5}$. Substitution of these roots into our general solution as given above yields:
$$y(x)=C_1e^{\frac{-1}{2}x}+C_2e^{\frac{2}{5}x}$$
the undetermined coefficients to our general condition, and therefore the particular solution, can be found from initial or boundary conditions to our differential equation. It is also worth mentioning that it is common to have complex roots and therefore solutions.
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