Question

The equation of the curve passing through the point $\left( a,–\frac{1}{a} \right)$ and satisfying the differential equation y – x$\frac{dy}{dx}$ = a $\left( y^{2} + \frac{dy}{dx} \right)$is -

• A. (x + a) (1 + ay) = – 4a²y
• B. (x + a) (1 – ay) = 4a²y
• C. (x + a) (1 – ay) = – 4a²y
• D. None of these

Answer 1

Answer: (C)

(x + a) (1 – ay) = – 4a²y

Answer 2

We have, y – x $\frac{dy}{dx}$ = a$\left( y^{2} + \frac{dy}{dx} \right)$

Ž y dx – x dy = ay² dx + ady

Ž y (1 – ay) dx = (x + a)dy Ž $\frac{dx}{x + a} - \frac{dy}{y(1 - ay)}$ = 0

Integrating, we get

log (x + a) – log y + log (1 – ay) = log c

log $\frac{(a + x)(1 - ay)}{y}$ = log c Ž (x + a)

(1 – ay) = cy.

Since the curve passes through $\left( a, - \frac{1}{a} \right)$,

2a × (1 + 1) = – $\frac{c}{a}$ Ž c = –4a².

So, the equation of curve is

(x + a) (1 – ay) = –4a²y.

Hence (3) is the correct answer