Variation of Parameters

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y₁(t) and y₂(t) are a fundamental set of solutions.

find a pair of functions, u₁(t) and u₂(t) so that

Here’s the assumption.  assume that whatever u₁(t) and u₂(t) are they will satisfy the following.

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{\displaystyle y^{(n)}(x)+\sum _{i=0}^{n-1}a_{i}(x)y^{(i)}(x)=b(x).\quad \quad {\rm {(i)}}}

Let {\displaystyle y_{1}(x),\ldots ,y_{n}(x)} be a fundamental system of solutions of the corresponding homogeneous equation

{\displaystyle y^{(n)}(x)+\sum _{i=0}^{n-1}a_{i}(x)y^{(i)}(x)=0.\quad \quad {\rm {(ii)}}}

Then a particular solution to the non-homogeneous equation is given by

{\displaystyle y_{p}(x)=\sum _{i=1}^{n}c_{i}(x)y_{i}(x)\quad \quad {\rm {(iii)}}}

The particular solution to the non-homogeneous equation can then be written as

{\displaystyle \sum _{i=1}^{n}y_{i}(x)\,\int {\frac {W_{i}(x)}{W(x)}}\ \mathrm {d} x.}