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Solving Linear Differential Equations of First Order

Solve: dydx...

Question

Solve: $\frac{d y}{d x} = e^{x} - y$ .

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Solution

$\frac{d y}{d x} = e^{x} - y$
or, $\frac{d y}{d x} + y = e^{x}$
or, $e^{x} \frac{d y}{d x} + e^{x} y = e^{2 x}$ [Multiplying both sides with $e^{x}$ ]
or, $\frac{d}{d x} (y e^{x}) = e^{2 x}$
Integrating we have
$y e^{x} = \frac{e^{2 x}}{2} + c$ [ $c$ being integrating constant]
or, $y = \frac{e^{x}}{2} + c e^{- x}$

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