# Solution to Linear First Order Ordinary Differential Equation/Solution by Integrating Factor

< Solution to Linear First Order Ordinary Differential Equation(Redirected from Solution by Integrating Factor)

## Proof Technique

The technique to solve a linear first order ordinary differential equation in the form:

- $\dfrac {\d y} {\d x} + P \left({x}\right) y = Q \left({x}\right)$

It immediately follows from Integrating Factor for First Order ODE that:

- $e^{\int P \left({x}\right) \rd x}$

is an integrating factor for $(1)$.

Multiplying it by:

- $e^{\int P \left({x}\right) \rd x}$

to reduce it to a form:

- $\dfrac {\d y} {\d x} e^{\int P \left({x}\right) \rd x} y = e^{\int P \left({x}\right) \rd x} Q \left({x}\right)$

is known as **Solution by Integrating Factor**.

It is remembered by the procedure:

*Multiply by $e^{\int P \left({x}\right) \rd x}$ and integrate.*

## Sources

- 1972: George F. Simmons:
*Differential Equations*... (previous) ... (next): $\S 2.10$