Answer to Question #185656 in Differential Equations for Haris khan

Question

Find the particular solution of nonlinear differential equation 

𝑑𝑦

𝑑𝑥

=

𝑦

2−1

𝑥

, 𝑦(1) = 2.

Answer 1

$\displaystyle y' = \frac{y^2 - 1}{x}\\ \frac{\mathrm{d}y}{y^2 - 1} = \frac{\mathrm{d}x}{x}\\ \int\frac{\mathrm{d}y}{y^2 - 1} = \int\frac{\mathrm{d}x}{x}\\ \ln\left(\frac{y - 1}{y + 1}\right) = 2\ln{x} + C\\ \frac{y - 1}{y + 1} = Ax^2\\ y - 1 = Ax^2(y + 1)\\ y(1 - Ax^2) = Ax^2 + 1\\ y = \frac{Ax^2 + 1}{1 - Ax^2}\\ y(1) = \frac{1 + A}{1 - A} = 2\\ 1 + A = 2 - 2A\\ 3A = 1, A = \frac{1}{3}\\ \therefore y = \frac{x^2 + 3}{3 - x^2}$