Second order linear ODE

Second order linear ODEs are in the following form:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y''+p(t)y'+q(t)y=g(t) }

Important types of second order linear ODEs include

• Homogeneous
• Constant coefficients (where p and q are constants)

Initial value problem

There are two arbitrary constants in the solution of a second order linear ODE, so we need two initial conditions.

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{cases} y(t_0)=y_0\\ y'(t_0)=y_0' \end{cases} }

Solutions

Constant coefficient, homogeneous

These are the simplest kind. They have the general form

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ay''+by'+cy=0 }

An exponential function has the property of being the same after many differentiation. We take advantage of this property and guess the solution to be the form of

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y=e^{rt} }

We substitute in the guess and obtain the characteristic equation

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{aligned} ar^2e^{rt}+bre^{rt}+ce^{rt}&=0\\ ar^2+br+c&=0 \end{aligned} }