Question

The solution of y = x $\frac{dy}{dx}$ + $\frac{dy}{dx}$ – $\left( \frac{dy}{dx} \right)^{2}$, is –

• A. y = (x – 1)²
• B. 4y = (x + 1)²
• C. (y – 1)² = 4x
• D. None of these

Answer 1

Answer: (B)

4y = (x + 1)²

Answer 2

The given equation can be written as;

y = xp + p – p² ; where p = $\frac{dy}{dx}$... (i)

differentiating both sides w.r.t. x, we get

p = p + $\frac{xdp}{dx}$ + $\frac{dp}{dx}$ – 2p $\frac{dp}{dx}$.

\ $\frac{dp}{dx}$ (x + 1 – 2p) = 0

\ either $\frac{dp}{dx}$ = 0, i.e., p = c ...(ii)

or x + 1 – 2p = 0, i.e., p = $\frac{1}{2}$ (x + 1) ... (iii)

Eliminating p between (i) and (ii) we get.

y = cx + c – c² as the complete solution and eliminating p between (i) and (iii)

y = $\frac{1}{2}$ (x + 1) x + $\frac{1}{2}$ (x + 1) – $\frac{1}{4}$ (x + 1)²

i.e., 4y = (x + 1)² as the singular solution.

Hence (2) is the correct answer