Question

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

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

Answer 1

Answer: (B)

(x + a) (1 – ay) = cy

Answer 2

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

Ž y – ay² = a$\frac{dy}{dx}$ + x $\frac{dy}{dx}$

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

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

On integrating both sides, we get

$\int_{}^{}\frac{dx}{(a + x)}$ = $\int_{}^{}\frac{dy}{y(1–ay)}$

Ž log (a + x) = $\int_{}^{}\left\lbrack \frac{1}{y} + \frac{a}{(1 - ay)} \right\rbrack$dx

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

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

Ž log (x + a) (1 – ay) = log cy

Ž (x + a) (1 – ay) = cy