Solution by Integrating Factor/Examples/y' + y = x^-1

Theorem
Consider the linear first order ODE:
 * $(1): \quad \dfrac {\d y} {\d x} + y = \dfrac 1 x$

with the initial condition $\tuple {1, 0}$.

This has the particular solution:
 * $y = \displaystyle e^{-x} \int_1^x \dfrac {e^\xi \rd \xi} \xi$

Proof
This is a linear first order ODE with constant coefficents in the form:
 * $\dfrac {\d y} {\d x} + a y = \map Q x$

where:
 * $a = 1$
 * $\map Q x = \dfrac 1 x$

with the initial condition $y = 0$ when $x = 1$.

Thus from Solution to Linear First Order ODE with Constant Coefficients with Initial Condition:

From Primitive of $\dfrac {e^x} x$ has no Solution in Elementary Functions, further work on this is not trivial.

Also see

 * Primitive of Exponential of a x over x