Reduction of order

Reduction of order is used to find the general solution of homogeneous linear second-order ODEs. The idea is that we can reduce a linear second-order ODE to a linear first-order ODE if we know one of the solutions $y_1$ (i.e., given $y_1$, we can find some linearly independent $y_2$). We want our ODE to be in standard form:

$$y^{\prime \prime }+P(x)y^{\prime }+Q(x)y=0$$

So by theorem, if $y_1(x)$ is a solution of the homogeneous ODE on an interval $I$, then:

$$y_2(x)=y_1(x)\int \frac{e^{-\int P(x)dx}}{y_1^2(x)}dx$$

Then, we use variation of parameters to find the general solution.

Derivation

Given one solution $y_1(x)$, we can represent our second solution with $y_2(x)=u(x)y_1(x)$. Then $y_2^{\prime }(x)=u(x)y_1^{\prime }(x)+u^{\prime }(x)y(x)$ and we also find $y_2^{\prime \prime }$. Substituting this into the left-side of the ODE, we get some expression that is a differential equation in terms of $u$.

The problems we get usually have the regular $u$ term disappear. So we get a second-order ODE with a second and first derivative term of $u$.

We can then substitute $w=u^{\prime }$ to get a first-order linear equation in terms of $w$. Using the integrating factor, we can solve this pretty easily to find $w=u^{\prime }$, then we can integrate $u^{\prime }$ to get $u$.

In this assumption, $y=u(x)\cdot y_1(x)$.