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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.


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}


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.