Answer to Question #154124 in Differential Equations for awais

Question

Solve the following Bernoulli’s equation: 𝑑𝑥 /𝑑𝑦 = 𝑥² + 2𝑥𝑦 + 𝑦²

Answer 1

$\displaystyle \frac{\mathrm{d}x}{\mathrm{d}y} = x^2 + 2xy + y^2\\ \frac{\mathrm{d}x}{\mathrm{d}y} = (x + y)^2\\ \frac{\mathrm{d}y}{\mathrm{d}x} = \frac{1}{(x + y)^2}\\ (x + y)^2\frac{\mathrm{d}y}{\mathrm{d}x} = 1\\ \frac{(x + y)^2}{1 + (x + y)^2}\frac{\mathrm{d}y}{\mathrm{d}x} = \frac{1}{1 + (x + y)^2}\\ \frac{1 + (x + y)^2 - 1}{1 + (x + y)^2}\frac{\mathrm{d}y}{\mathrm{d}x} = \frac{1}{1 + (x + y)^2}\\ \frac{\mathrm{d}y}{\mathrm{d}x} - \frac{1}{1 + (x + y)^2}\frac{\mathrm{d}y}{\mathrm{d}x} = \frac{1}{1 + (x + y)^2}\\ \frac{\mathrm{d}y}{\mathrm{d}x} = \frac{1}{1 + (x + y)^2}\left(1 + \frac{\mathrm{d}y}{\mathrm{d}x}\right)\\ \frac{\mathrm{d}y}{\mathrm{d}x} = \frac{1}{1 + (x + y)^2}\cdot\frac{\mathrm{d}}{\mathrm{d}x}\left(x + y\right)\\ \frac{\mathrm{d}y}{\mathrm{d}x} = \frac{\mathrm{d}}{\mathrm{d}x}\left(\arctan(x + y)\right)\\ \int\frac{\mathrm{d}y}{\mathrm{d}x} \cdot \mathrm{d}x= \int \frac{\mathrm{d}}{\mathrm{d}x}\left(\arctan(x + y)\right) \cdot \mathrm{d}x\\ y = \arctan(x + y) + C$