Differential equation

Question

Consider x ≠ 0. Solve the differential equation

2ydy/dx = 5+2x-2y²

Determine the particular solution that passes through the point (0, √ 5) and the maximum range where it is defined.

Answer 1

2ydy/dx + 2y² = 5 - 2x; or dy/dx + y = (2.5 - x)y^-1.

This is Bernully equation. We use substitution: u = y²; u' = 2yy'.

u' + 2u = 5 - 2x. Integrated factor M = e^∫2dx = e^2x;

e^2x(u' + 2u) = (5 - 2x)e^2x; (e^2xu)^' = (5 - 2x)e^2x; e^2xu = ∫(5 - 2x)e^2x; e^2xu = (5 -2x)e^2x/2 + e^2x/2 + C

u = 3 - x + Ce^-2x; y = ± [3 - x + Ce^-2x)^1/2