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

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## Contents

## 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:

\(\displaystyle y\) | \(=\) | \(\displaystyle e^{-x} \int_1^x \dfrac {e^\xi \rd \xi} \xi + 0 \cdot e^{x - 1}\) | |||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle e^{-x} \int_1^x \dfrac {e^\xi \rd \xi} \xi\) |

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

$\blacksquare$

## Also see

## Sources

- 1958: G.E.H. Reuter:
*Elementary Differential Equations & Operators*... (previous) ... (next): Chapter $1$: Linear Differential Equations with Constant Coefficients: $\S 1$. The first order equation: $\S 1.2$ The integrating factor:*Example*$2$