Complex Math Simple Life

I Order Differential Equations Part II

Posted in differential, equation by beauangel on the February 17, 2007

Linear Differential Equations

Standard Form:

\frac{dy}{dx} + p\:y = Q(x) where p and Q are functions of x.

To solve this type of differential equation, we multiply the equation throughout by the factor e^{\int p dx} called the Integrating Factor and hence convert the LHS into an exact differential.

Multiplying throughout by e^{\int p dx}, we get

e^{\int p dx}. \frac{dy}{dx} + e^{\int p dx}.p\:y = Q\:e^{\int p dx}

which becomes

\frac{d}{dx} \left( e^{\int p dx}y \right) = Qe^{\int p dx}

Integrating both sides w.r.t x, we get

\int \frac{d}{dx} = \int Q e^{\int p dx}dx

ie, e^{\int p dx}y = \int Q e^{\int p dx}dx

Therefore, the formula for the General Solution of a Linear Differential Equation is:

ye^{\int p dx} = \int Q e^{\int p dx}dx

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