Solve the differential equation: (x + 1)dy dx -y = e^ 3x (x + 1)^… - Vidyadip

Question

Solve the differential equation: $(x + 1)\frac{dy}{dx} -y = e^{3x} (x + 1)^{3}$

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Answer

The given differential equation can be written as
$\frac{dy}{dx} - \frac{1}{x + 1} y = (x + 1) . e^{3x}$
Here, integrating factor = $e^{\int - \frac{1}{x + 1} dx} = \frac{1}{x + 1}$
$\therefore \text{Solution is} \frac{1}{x + 1} = \int(x + 1) e^{3x} dx$
$\therefore \frac{y}{x + 1} = (x + 1)\frac{e^{3x}}{3} - \frac{e^{3x}}{9} + \text{C}$
or $\text y = \bigg[\frac{1}{3}(x + 1)^{2} - \frac{x + 1}{9}\bigg]e^{3x} + \text{C }(x + 1)$

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