Second Order Differential Equations

Linear Second-Order Differential Equations

Second order differential equations consist of second-order derivatives and require 2 initial conditions $y(0)=y\_0\text{and}{y}^{\prime }(0)=y^{\prime }\_0$ to be solved. Their typical structure is

$$P(x)y^{\prime \prime }(x)+Q(x)y^{\prime }(x)+R(x)y(x)=G(x)$$

If $G(x)=0$ the equation will be considered homogenous.

Superposition Solution Principle

If it is true that the equation we’re dealing with is homogenous, the following Theorem holds.

If two linearly independent solutions (not constant multiples of eachother) of the differential equation $y\_1$ and $y\_2$ are known, then for any constants $c\_1$ and $c\_2$ the linear combination $y=c\_1y\_1(x)+c\_2y\_2(x)$ is also a solution to the equation. The Auxiliary Equation arises from that and provides solutions to the differential equation.

$$y(x)=c\_1y\_1(x)+c\_2y\_2(x)$$

Nonhomogenous Linear Differential Equations

Analogies of Electrical Circuits and Mechanical Motion modeled by Differential Equations