# First Order Linear Differential Equation

$$\frac{dy}{dx} = p \left( x \right) y + q \left( x \right)$$

Procedure:

– Calculate $I = \int p\left( x \right) \, dx$

– Find the integrating factors $e^{-I \left( x \right)}$ and $e^{I \left( x \right)}$

– Evaluate $\int e^{I \left( x \right)} q \left( x \right) \, dx$

– Hence the solution:

$$y = e^{-I \left( x \right)} \left( \int e^{I \left( x \right)} q \left( x \right) \, dx \right) + C e^{-I \left( x \right)}$$

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