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JEE Previous Year Question

$The general solution of the differential equation \frac{dy}{dx} + \frac{y}{x} = x^{2} is$

Topic: JEE PYQ

Answer: C. Full step-by-step solution below.

A $xy = \frac{x ^{2}}{2} + c$
B $xy = \frac{x ^{3}}{3} + c$
C $xy = \frac{x ^{4}}{4} + c$
D $xy = \frac{x ^{5}}{5} + c$

Correct answer: C

Solution

(c) \frac{dy}{dx} + \frac{y}{x} = x^{2} \Rightarrow P = \frac{1}{x}, Q = x^{2} \Rightarrow IF = e^{\int Pdx} = e^{\int \frac{1}{x} dx} = e^{ln x} = x \Rightarrow y \cdot IF = \int Q \cdot IFdx + C \Rightarrow xy = \int x^{2} \cdot xdx + C ∴ xy = \frac{x ^{4}}{4} + C .

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