User:MFH/math/variation of constant

More concretely, in the case of linear differential equations of order n, it can be shown that taking the constants appearing in the homogenious solution as undetermined functions, one obtains a system of differential equations of order n-1 for these functions, which allows eventually to find the general solution to the initial equation.

Example 2

To solve the first order linear differential equation

$y^{\prime }(x)+g(x)y(x)=f(x)$

first consider the homogenious equation (f=o), to find $y(x)=C{e}^{(}-G(x)),G(x)=\int g(x)dx$. Then replace the constant $C$ with a function $C(x)$.