Linear Equations | Equations of Order One

Linear Equations of Order One
Linear equation of order one is in the form
 

$\dfrac{dy}{dx} + P(x) \, y = Q(x).$

 

The general solution of equation in this form is
 

$\displaystyle ye^{\int P\,dx} = \int Qe^{\int P\,dx}\,dx + C$

 

Derivation
$\dfrac{dy}{dx} + Py = Q$
 

Use $\,e^{\int P\,dx}\,$ as integrating factor.
$e^{\int P\,dx} \dfrac{dy}{dx} + Pe^{\int P\,dx} \, y = Qe^{\int P\,dx}$
 

Multiply both sides of the equation by dx
$e^{\int P\,dx} \,dy + Pe^{\int P\,dx} \, y \, dx = Qe^{\int P\,dx}\, dx$
 

Let
$u = \int P\,dx$

$du = P\,dx$
 

Thus,
$e^u \,dy + ye^u \, du = Qe^u\, dx$

$d(e^u y) = Qe^u\, dx$

$\displaystyle d(e^u y) = \int Q e^u\, dx$
 

But $\,u = \int P\,dx\,$. Thus,

$\displaystyle ye^{\int P\,dx} = \int Qe^{\int P\,dx}\,dx + C$

 

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