# 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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