QJEEM21Maed38: JEE Main 2021 Mathematics - Class 12 JEEM21Maed38 Solution: Let y = y(x) be the solution of the differential equation (x x…

Question

Verified Solution • 2025–26 Session moderate

Let y = y(x) be the solution of the differential

equation (x

-

x³)dy = (y + yx²

-

3x⁴)dx, x > 2. If y(3) = 3, then y(4) is equal to :

Accepted Answer

(x - {x^3})dy = (y + y{x^2} - 3{x^4})dx

\Rightarrow xdy - ydx = (y{x^2} - 3{x^4})dx + {x^3}dy

\Rightarrow {{xdy - ydx} \over {{x^2}}} = (ydx + xdy) - 3{x^2}dx

\Rightarrow d\left( {{y \over x}} \right) = d(xy) - d({x^3})

Integrate

\Rightarrow {y \over x} = xy - {x^3} + c

given f(3) = 3

\Rightarrow {3 \over 3} = 3 \times 3 - {3^3} + c

\Rightarrow c = 19

\therefore

{y \over x} = xy - {x^3} + 19

at

x = 4,{y \over 4} = 4y - 64 + 19

15y = 4 \times 45

\Rightarrow y = 12

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