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UPSEE 2019 Solved Paper

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Question : 124 of 150

Marks: +1, -0

If

y(x)

satisfies equations

(1+x^{2})\frac{dy}{dx}+

2xy-4x^{2}=0

y(0)=0

y(1)

is

$\frac{2}{3}$
1
$\frac{4}{3}$
$\frac{5}{3}$

Solution:

We have,

(1+x^{2})\frac{dy}{dx}+

2xy-4x^{2}=0

⇒

\frac{dy}{dx}+\frac{2xy}{1+x^{2}}=\frac{4x^{2}}{1+x^{2}}

This is linear differential equation

\therefore I.F=e^{\int\frac{2x}{1+x^{2}}\,dx}=e^{\log(1+x^{2})}

=1+x^{2}

Solution of given differential equation is

y(1+x^{2})=\int 4x^{2}\,dx

y(1+x^{2})=\frac{4x^{3}}{3}+C

put

x=0, y=0,

we get

C=0

\therefore y(1+x^{2})=\frac{4x^{3}}{3}

put

x=1, y=\frac{2}{3}

\therefore y(1)=\frac{2}{3}

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