Final answer to the problem
$y=\sqrt{6\left(\ln\left(x\right)+c_0\right)}x,\:y=-\sqrt{6\left(\ln\left(x\right)+c_0\right)}x$
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Step-by-step Solution
Learn how to solve integrals of polynomial functions problems step by step online. Solve the differential equation dy/dx=(3x^2+y^2)/(xy). We can identify that the differential equation \frac{dy}{dx}=\frac{3x^2+y^2}{xy} 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: y=ux. Expand and simplify. Integrate both sides of the differential equation, the left side with respect to u, and the right side with respect to x.
Final answer to the problem
$y=\sqrt{6\left(\ln\left(x\right)+c_0\right)}x,\:y=-\sqrt{6\left(\ln\left(x\right)+c_0\right)}x$
