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Solving Linear Differential Equations of First Order

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Question

Find the solution to the differential equation:
$\frac{d y}{d x} + 6 y = 18 e^{3 x}$ .

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Solution

Given,
$\frac{d y}{d x} + 6 y = 18 e^{3 x}$
or, $e^{6 x} \frac{d y}{d x} + 6 e^{6 x} y = 18 e^{9 x}$ [ Since the integrating factor is $e^{6 x}$ ]
or, $\frac{d}{d x} (y e^{6 x}) = 18 e^{9 x}$
Now integrating both sides we get,
$y e^{6 x} = 2 e^{9 x} + c$ [ Where $c$ is integrating constant]

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