The solution of differential equation xdy/dx + 2y = x² { is}

Correct Answer is : y=x4+C4x2y = {x^{4}+C}{4 x²}y=4x2x4+C

Correct Answer is : y=x4+C4x2y = {x^{4}+C}{4 x²}y=4x2x4+C

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KCET 2015 Math Solved Paper

Question : 54 of 60

Marks: +1, -0

x\frac{dy}{dx} + 2y = x^2 \text{ is}

Solution:

Given differential equations,

x \; \frac{dy}{dx} + 2y = x^{2}

\Rightarrow \; \frac{dy}{dx} + \; \frac{2y}{x} = x

→ (1)

General differential equation of this type is given by

\frac{dy}{dx} + P(x)y = Q(x)

So,

P = \frac{2}{x}

then I.F. factor is given by

e^{\int Pdx} = e^{\int \frac{2}{x} dx} = e^{2 \log x} = e^{\log x^{2}} = x^{2}

Then,

x^{2} \; \frac{dy}{dx} + 2xy = x^{3}

\Rightarrow \; \frac{d}{dx}(x^{2} y) = x^{3}

Integrating both the sides, we get

x^{2} y = \; \frac{x^{4}}{4} + C_{1}

\Rightarrow x^{2} y = \; \frac{x^{4}+C}{4}

\Rightarrow y = \; \frac{x^{4}+C}{4 x^{2}}

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