Second-order non-homogeneous differential equations – A Level Further Mathematics Edexcel Revision

Second-order non-homogeneous differential equations are a type of differential equation where the highest derivative is second order and the equation is not homogenous.
A second-order non-homogeneous differential equation has the general form: d²y/dx² + p(x)dy/dx + q(x)y = r(x), where p(x), q(x), and r(x) are any given functions of x, and r(x) ≠ 0.
In such an equation, ‘r(x)’ can be a function of ‘x’ or even a constant. If r(x) were equal to zero, then we would have a homogeneous equation, however this is not the case for a non-homogeneous equation.
The solutions of non-homogeneous differential equations consist of two parts: Complementary function (CF) and Particular Integral (PI).
The complementary function is the general solution of the corresponding homogeneous equation: d²y/dx² + p(x)dy/dx + q(x)y = 0.
The Particular Integral (PI) is any specific solution to the non-homogeneous differential equation.
To find the PI, one technique can be trying the method of undetermined coefficients. Another, often simpler technique, is to use variation of parameters.
By adding the CF and the PI, a general solution of the non-homogeneous equation can be obtained.
The initial conditions of the problem can then be used to find the exact solution by replacing the constant ‘C’ in the general solution.
To fully understand and master this topic, it’s crucial to work on a variety of problem sets, including those with different types of right-hand side functions r(x).
Mistakes can be avoided by ensuring correct notation and always carefully checking intermediate and final answers.
The ability to solve these kinds of equations is vital for many areas of mathematics and its applications, including physics and engineering.

Do note that this recap is only part of a much bigger picture. Attempt numerous solved examples and exercises to develop a robust understanding of the concepts and techniques.