Second Order ODE/y'' - x f(x) y' + f(x) y = 0

Theorem
The second order ODE:
 * $(1): \quad y'' - x f \left({x}\right) y' + f \left({x}\right) y = 0$

has the solution:
 * $\displaystyle y = C_1 x + C_2 x \int x^{-2} e^{\int x f \left({x}\right) \, \mathrm d x} \, \mathrm d x$

Proof
Note that:

Substituting into $(1)$:

and so it has been demonstrated that:
 * $y_1 = x$

is a particular solution of $(1)$.

$(1)$ is in the form:
 * $y'' + P \left({x}\right) y' + Q \left({x}\right) y = 0$

where:
 * $P \left({x}\right) = -x f \left({x}\right)$

From Particular Solution to Homogeneous Linear Second Order ODE gives rise to Another:
 * $y_2 \left({x}\right) = v \left({x}\right) y_1 \left({x}\right)$

where:
 * $\displaystyle v = \int \dfrac 1 { {y_1}^2} e^{-\int P \, \mathrm d x} \, \mathrm d x$

is also a particular solution of $(1)$.

We have that:

Hence:

and so:

From Two Linearly Independent Solutions of Homogeneous Linear Second Order ODE generate General Solution:


 * $\displaystyle y = C_1 x + C_2 x \int x^{-2} e^{\int x f \left({x}\right) \, \mathrm d x} \, \mathrm d x$