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

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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