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

“Second-order homogeneous differential equations” involve equations that equate to zero and include second-order derivatives.
To differentiate an equation means to find the rate at which the original function changes. In the case of second-order derivatives, you’re finding the rate at which the rate changes.
These types of equations are important in various fields, notably in physics where they can describe motion or other physical properties changing with respect to time.
They have a general form, often written as ay’’ + by’ + c*y = 0. Here, ‘y’ is the dependent variable, ‘a’, ‘b’ and ‘c’ are constants, and y’’ is the second derivative of y with respect to the independent variable, while y’ is the first derivative of y.
The solutions to these second-order homogeneous differential equations is often searched in the form of y = e^(r*x). By substituting this solution form into the differential equation, a characteristic equation arises.
The characteristic equation is a quadratic equation of the form ar^2 + br + c = 0 where roots determine the nature of the solution.
There are three possible types of solutions based on the discriminant (B^2 - 4AC) values in the characteristic equation: Real and different roots, Real and same roots, and Complex roots.
For real and different roots, the general solution will be y = Ae^(r1x) + Be^(r2x), where A and B are arbitrary constants and r1, r2 are the roots of the characteristic equation.
For real and same roots (repeated root), the general solution will be y = (A + B * x) e^(r * x).
For complex roots, the general solution will be y = e^(ax)(Ccos(bx) + D*sin(bx)), where ‘a’ is the real part and ‘b’ is the imaginary part.
To further define the exact solution, initial conditions (i.e, values of the original function and its first derivative at specific points) are normally given.