Final answer to the problem
$\frac{\sqrt{y}\arctan\left(\frac{x}{\sqrt{\left(-x+y\right)y}}\right)}{\sqrt{-x+y}}=\ln\left|y\right|+C_0$
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Step-by-step Solution
Learn how to solve problems step by step online. Solve the differential equation dy/dx=(y^2)/(x^2+y^2). We can identify that the differential equation \frac{dy}{dx}=\frac{y^2}{x^2+y^2} is homogeneous, since it is written in the standard form \frac{dy}{dx}=\frac{M(x,y)}{N(x,y)}, where M(x,y) and N(x,y) are the partial derivatives of a two-variable function f(x,y) and both are homogeneous functions of the same degree. Use the substitution: x=uy. Expand and simplify. Integrate both sides of the differential equation, the left side with respect to u, and the right side with respect to y.
Final answer to the problem
$\frac{\sqrt{y}\arctan\left(\frac{x}{\sqrt{\left(-x+y\right)y}}\right)}{\sqrt{-x+y}}=\ln\left|y\right|+C_0$
