NDA 2023 Maths · Mathematics · Differential Equations · hard

If xdy/dx+2y=x^2 for x>0 and y(1)=1, then y(2) is equal to

Single correct option

x\dfrac{dy}{dx}+2y=x^2

for

x>0

and

y(1)=1

, then

y(2)

is equal to

Show answer and solution

Answer: (B) $\dfrac{19}{16}$

The equation is

\dfrac{dy}{dx}+\dfrac2xy=x

with integrating factor

x^2

, so

\dfrac{d}{dx}(x^2y)=x^3

and

x^2y=\dfrac{x^4}4+C

. Using

y(1)=1

gives

C=\dfrac34

. At

x=2

4y=4+\dfrac34

, i.e.

y=\dfrac{19}{16}

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