A linear ODE with constant coefficients looks like this:
$$a_n\frac{\mathrm{d}^ny}{\mathrm{d}x^n} + a_{n-1}\frac{\mathrm{d}^{n-1}y}{\mathrm{d}x^{n-1}} + … + a_1\frac{\mathrm{d}y}{\mathrm{d}x} + a_0y = f(x)$$
For example:
$$\frac{\mathrm{d}^2y}{\mathrm{d}x^2} – 6\frac{\mathrm{d}y}{\mathrm{d}x} + 13y = x^3$$
This is the most common form of differential equation in circuit theory.
Note that if the coefficients are not constant (i.e. a function of x) then most analytical methods do not work.
Continue reading “Linear Differential Equations with Constant Coefficients”