Final answer to the problem
$-\ln\left(-2\left(\frac{y}{x}\right)^2+1\right)+2\ln\left(\frac{y}{x}\right)=\ln\left(x\right)+C_0$
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Step-by-step Solution
Learn how to solve integration by trigonometric substitution problems step by step online. Solve the differential equation (2y^3-3yx^2)dx+2x^3dy=0. We can identify that the differential equation \left(2y^3-3yx^2\right)dx+2x^3dy=0 is homogeneous, since it is written in the standard form M(x,y)dx+N(x,y)dy=0, 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
$-\ln\left(-2\left(\frac{y}{x}\right)^2+1\right)+2\ln\left(\frac{y}{x}\right)=\ln\left(x\right)+C_0$
