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In each of the following differential equation show that it is homogeneous and solve it.

contexto.140

Solution:

\begin{array}{l} \Rightarrow\left(x^{3}+3 x y^{2}\right) d x+\left(y^{3}+3 x^{2} y\right) d y=0 \\ \left.\left.\Rightarrow \frac{d y}{d x}=-\frac{\left(x^{3}+3 x y^{2}\right)}{\left(y^{2}+3 x^{2} y\right)}=-\frac{3 x y^{2}\left(\frac{x^{3}}{3 x y^{2}}+1\right)}{3 x^{2} y} \frac{y^{3}}{3 x^{2} y}+1\right)=-\frac{y\left(\frac{x^{2}}{3 y^{2}}+1\right)}{x} \frac{y^{2}}{3 x^{2}}+1\right) \\ \Rightarrow \frac{d y}{d x}=f\left(\frac{y}{x}\right) \end{array}
the given differential eq. is a homogenous eq.
Solution of the given differential eq. is:

On integrating both sides we obtain:

By resubstituting the value of we have