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:

$\dfrac {\d y} {\d x} + \map P x y = \map Q x$


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

$e^{\int \map P x \rd x}$

is an integrating factor for $(1)$.


Multiplying it by:

$e^{\int \map P x \rd x}$

to reduce it to a form:

$\dfrac {\d y} {\d x} e^{\int \map P x \rd x} y = e^{\int \map P x \rd x} \map Q x$

is known as Solution by Integrating Factor.


It is remembered by the procedure:

Multiply by $e^{\int \map P x \rd x}$ and integrate.


Examples

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

The linear first order ODE:

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

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:

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


Sources