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JEE Main 2016

JEE Main 2016 Mathematics question on Differential Equations

If a curve y=f(x)y = f(x) passes through the point (1,−1)(1, -1) and satisfies the differential equation, y(1+xy)dx=x dyy(1 + xy)dx = x\,dy, then f(−12)f\left(-\frac{1}{2}\right) is equal to
  1. A.
    −25-\dfrac{2}{5}
  2. B.
    −45-\dfrac{4}{5}
  3. C.
    25\dfrac{2}{5}
  4. D.
    45\dfrac{4}{5}
Show answer and solution

Correct option: D

Solution

Rewrite y(1+xy)dx = x dy as y dx + xy²dx = x dy, i.e., (y dx - x dy)/y² = -x dx, which is d(x/y) = -x dx.

Integrating: x/y = -x²/2 + C.

Apply (1,-1): 1/(-1) = -1/2 + C → C = -1/2.

So x/y = -x²/2 - 1/2, giving y = x/(-x²/2 - 1/2) = -2x/(x²+1).

At x = -1/2: y = -2(-1/2)/((1/4)+1) = 1/(5/4) = 4/5.

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