Find the equation of the curve which passes through the point (2, 2) and satisfies the differential equation y - x(dy/dx) = y^2 + dy/dx.

Question

Find the equation of the curve which passes through the point (2, 2) and satisfies the differential equation

\(y-x\frac{dy}{dx}=y^2+\frac{dy}{dx}.\)

Answer 1

Given the differential equation

Integrating both sides we have,

⇒ log|y| + log|1 – y| = log|1 + x| + logc

⇒ log|y(1 – y)| = log|c(1 + x)|

⇒ y(1 – y) = c(1 + x) ……(1)

Since, the equation passes through (2, 2), So,

2(1 – 2) = c(1 + 2)

⇒ – 2 = c × 3

\(\Rightarrow c = -\frac{2}{3}\)

Therefore, equation (1) becomes

y(1 – y) = \(-\frac{2}{3}\)(1 + x)