Second Order Linear ODE

A linear second order ODE is a differential equation that is express as a linear combination of a function and it’s derivatives.

The general form of a linear second order ODE is:

$$\frac{d^{2}y}{d{x}^2}+f(x)\frac{dy}{dx}+g(x)y=h(x)$$

• If $h(x)=0$, the equation is homogenous.
• Else, the equation is said to be inhomogeneous

Warning

Homogenous Second Order ODEs are different from homogenous type First Order ODEs.

Theorem - Homogenous Equations §

The general solution of a homogenous equation $y”+f(x)y^{\prime }+g(x)y=0$ is:

$$y(x)=A{y}_1(x)+B{y}_2(x)$$

• $A,B$ are both constants
• $y_1,y_2$ are two linearly independent solutions of the ODE.

Inhomogeneous Solutions §

The full solution of the inhomogeneous equation $y^{\prime \prime }+f(x)y^{\prime }+g(x)y=h(x)$, when $h(x)\ne 0$ is:

$$y(x)=y_h(x)+y_p(x)$$

• $y_h(x)$ is the homogenous solution to the ODE
• $y_p(x)$ is the particular solution.

See Second Order Linear ODEs (Constant Coefficients) to solve equations with constant coefficients.