Linear Second Order ODE/(x - 1) y'' - y' + y = 0

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

has the general solution:
 * $y = C_1 x + C_2 e^x$

Proof
Note that:

and so by inspection:
 * $y_1 = x$

is a particular solution of $(1)$.

$(1)$ can be expressed as:
 * $(2): \quad y'' - \dfrac x {x - 1} y' + \dfrac 1 {x - 1} y = 0$

which is in the form:
 * $y'' + \map P x y' + \map Q x y = 0$

where:
 * $\map P x = -\dfrac x {x - 1}$

From Particular Solution to Homogeneous Linear Second Order ODE gives rise to Another:
 * $\map {y_2} x = \map v x \, \map {y_1} x$

where:
 * $\ds v = \int \dfrac 1 { {y_1}^2} e^{-\int P \rd x} \rd x$

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:


 * $y = C_1 x + C_2 e^x$