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 }

Particular solution

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} }

Depending on the constants, it will give us anywhere from zero to two solutions: 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_1} and 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_2}

General solution

For reasons that will be elaborated on in the future, the general solution is of 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(t) = c_1 y_1+c_2 y_2 }