# Solution by Integrating Factor/Examples

## Examples of Use of Solution by Integrating Factor

### Example: $y' - 3 y = \sin x$

$\dfrac {\d y} {\d x} - 3 y = \sin x$

has the general solution:

$y = \dfrac 1 {10} \paren {3 \sin x - \cos x} + C e^{3 x}$

### Example: $y' + y = \dfrac 1 x$

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:

$y = \ds e^{-x} \int_1^x \dfrac {e^\xi \rd \xi} \xi$