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We can identify that the differential equation $\frac{dy}{dx}=\frac{-y\left(2x^3-y^3\right)}{x\left(2y^3-x^3\right)}$ 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
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$\frac{dy}{dx}=\frac{-y\left(2x^3-y^3\right)}{x\left(2y^3-x^3\right)}$
Learn how to solve differential equations problems step by step online. Solve the differential equation dy/dx=(-y(2x^3-y^3))/(x(2y^3-x^3)). We can identify that the differential equation \frac{dy}{dx}=\frac{-y\left(2x^3-y^3\right)}{x\left(2y^3-x^3\right)} 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.