Differential Equations Question - None

Question

Find the solution of the differential equation x\frac{dy}{dx} + y - x + xy\cot x = 0, where x eq 0

Accepted Answer

Step 1: Given equation: x\frac{dy}{dx} + y - x + xy\cot x = 0

Step 2: Rearrange to get:

x\frac{dy}{dx} + y(1 + x\cot x) = x

Step 3: Divide both sides by x:

\frac{dy}{dx} + \frac{y(1 + x\cot x)}{x} = 1

Step 4: This is a linear differential equation of the form \frac{dy}{dx} + P(x)y = Q(x), where P(x) = \frac{1 + x\cot x}{x} and Q(x) = 1

Step 5: Find the integrating factor: \mu(x) = e^{\int P(x)dx}

Step 6: The solution is y = \frac{1}{x} - \cot x + \frac{C}{x\sin x}

∴ Answer is C.

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