Solve the differential equation x dy dx + y = x x + x, given that… - Vidyadip

Question

Solve the differential equation $x\frac{\text{dy}}{\text{d}x} + \text{y} = x \cos x + \sin x,$ given that y = 1 when $x = \frac{\pi}{2}.$

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Answer

The given equation can be written as
$\frac{\text{dy}}{\text{dx}} + \frac{\text{y}}{\text{x}} = \cos \text{x} + \frac{\sin \text{x}}{\text{x}}$
$\text{I.F.} = \text{e}^{\int \frac{1}{\text{x}} \text{dx}} = \text{e}^{\log \text{x}} = \text{x}$
$\therefore$ Solution is
$\text{y. x} = \int \text{(x} \cos \text{x} + \sin \text{x}) \text{dx + c}$
$\Rightarrow \text{x . y} = \text{x } \sin \text{x + c}$
$\text{or} \text{ y} = \sin \text{x} + \frac{\text{c}}{\text{x}}$
$\text{when x} = \frac{\pi}{2}, \text{y} = 1, \text{we get c = 0}$
Required solution is $\text{y} = \sin \text{x}$

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