# First Order Linear Differential Equation

In the page before we demonstrated that the first order linear differential equation :

$y' + P(x)y = Q(x)$

Has this solution :

$y = {1 \over {e^{\int {P(x)dx} } }}\left( {\int {Q(x)e^{\int {P(x)dx} } dx} + C} \right)$

In physics and especially in electronics we will find easily functions like that below :

$y' + ay = b$

this differential equation is a simpler case of our generic.

P(x)=a
Q(x)=b

We notice that

$\int {P(x)dx=\int {adx = ax}}+c_1 = ax$

C1 can be set to zero since we already have a constant C

$y = {1 \over {e^{ax} }}\left( {b\int {e^{ax} dx} + C} \right) = {1 \over {e^{ax} }}\left( {{{be^{ax} } \over a}+ C} \right) = {b \over a} + Ce^{ - ax}$

Resuming

$y'(x) + P(x)y(x) = Q(x) \Rightarrow y = {1 \over {e^{\int {P(x)dx} } }}\left( {\int {Q(x)e^{\int {P(x)dx} } dx} + C} \right)$ (A)

$y'(x) + ay(x) = b \Rightarrow y = {b \over a} + Ce^{ - ax}$ (B)

The (B) solution will be heavily used in this site since it is extremely useful.