Second Order Non-Homogeneous Differential Equations

Understanding the Equations

Second order non-homogeneous differential equations are a class of differential equations of the form d²y/dx² + p(x)dy/dx + q(x)y = r(x), where r(x) ≠ 0.
The left-hand side is a second order linear differential operator acting on y, while the right-hand side r(x) characterises these equations as non-homogeneous.
The general solution to these equations is the superposition of the general solution of the corresponding homogeneous equation and any particular solution to the original non-homogeneous equation.

In order to solve these kinds of differential equations, two broad strategies can be employed: method of undetermined coefficients (also known as method of educated guess) and the method of variation of parameters.
The method of undetermined coefficients involves guessing a form for the particular solution, plugging it back into the original differential equation to solve for the coefficients of the guess.
The method of variation of parameters assumes the particular solution to be a variation of the general solution’s parameters. This method is useful when r(x) is not of a form suitable for the method of undetermined coefficients.

The solutions to second order non-homogeneous differential equations can broadly be divided into three types based on roots of the auxiliary equation (m² + pm + q = 0): real and distinct roots, real and repeated roots, and complex roots.
In the real and distinct roots case, the general solution will be of the form y = Ae^^(m1x) + Be^^(m2x).
In the real and repeated roots case, the general solution will be of the form y = Ae^^(mx) + Bxe^^(mx).
In the complex roots case, the general solution will be of the form y = e^^(αx) [Acos(βx) + Bsin(βx)].
The particular solution is added to the relevant general solution to obtain the complete solution to the non-homogeneous differential equation.
These solutions offer rich information on the system’s future behaviour and often appear in various applications including control systems, electrical circuits, and mechanical systems illustrating the relevance of these differential equations.

Once a solution has been obtained, it’s important to verify the solution by substituting it back into the original differential equation.
As with any mathematical model, the accuracy and validity of the solutions should be compared with real-world data or expected outcomes. Deviations might demand refinement of the model.