Question

Solve the Differential Equation
xdx + ydy + (x²+y²)dy =0​

Answer 1

Step-by-step explanation:

$xdx + ydy + ( {x}^{2} + {y}^{2} )dy = 0 \\ \\ d(xy) = - ( {x}^{2} + {y}^{2} )dy \\ \\ d(xy) = - \frac{1}{ {y}^{2} } ( {x}^{2} {y}^{2} + 1)dy \\ \\ \frac{d(xy)}{1 + (xy) {}^{2} } = \frac{ - 1}{ {y}^{2} } dy$

Let xy =z and integrate both sides

$\int \frac{dz}{1 + {z}^{2} } = - \int \frac{1}{ {y}^{2} } dy \\ \\ \tan {}^{ - 1} (z) = \frac{1}{y} + c \\ \\ \tan {}^{ - 1} (xy) = \frac{1}{y} + c$