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If the right-hand side of the equation $\frac{dy}{dx}=f\left(x,y\right)$can be expressed as a function of the ratio $\frac{y}{x}$alone, then we say the equation is homogeneous.

Equations of the form$\frac{dy}{dx}=G\left(ax+by\right)$

When the right-hand side of the equation $\frac{dy}{dx}=f\left(x,y\right)$can be expressed as a function of the combination $ax+by$, where a and b are constants, that is,$\frac{dy}{dx}=G\left(ax+by\right)$then the substitution $z=ax+by$transforms the equation into a separable one.

A first-order equation that can be written in the form $\frac{dy}{dx}+P\left(x\right)y=Q\left(x\right)y^n$, where P9x) and Q(x) are continuous on an interval (a, b) and n is a real number, is called a Bernoulli equation.

We have used various substitutions for to transform the original equation into a new equation that we could solve. In some cases, we must transform bothandinto new variables, sayand. This is the situation for equations with linear coefficients-that is, equations of the form

.$\left(a_{1}x+b_{1}y+c_{1_{}}\right)dx+\left(a_{2}x+b_{2}y+c\right)dy=0$

Given, $\left({\mathbf{y}}^{\mathbf{3}}\mathbf{-}胃{\mathbf{y}}^{\mathbf{2}}\right)\mathbf{d}胃\mathbf{+}\mathbf{2}{胃}^{\mathbf{2}}\mathbf{ydy}\mathbf{=}\mathbf{0}$$\left(y^3-胃y^2\right)d胃+2{胃}^{2}ydy=0$

By evaluating

$\begin{array}{rcl}\left(y^3-胃y^2\right)d胃+2{胃}^{2}ydy& =& 0\\ 2{胃}^{2}ydy& =& -\left(y^3-胃y^2\right)d胃\\ \frac{dy}{d胃}& =& \frac{-y^3+胃y^2}{2{胃}^{2}y}\\ & =& -\frac{y^2}{2{胃}^2}+\frac{y}{2胃}\\ & =& -\frac{1}{2}\left(\frac{y}{胃}\right)+\frac{1}{2}\left(\frac{y}{胃}\right)\end{array}$

Let us consider $\mathbf{v}\mathbf{=}\frac{\mathbf{y}}{胃}$

Then,

role="math" localid="1655118354092" $\begin{array}{rcl}\frac{\mathbf{1}}{\mathbf{2}}\left(\mathbf{-}{\left(\frac{\mathbf{y}}{胃}\right)}^{\mathbf{2}}\mathbf{+}\frac{\mathbf{y}}{胃}\right)& \mathbf{=}& \mathbf{G}\left(\frac{\mathbf{y}}{胃}\right)\\ & \mathbf{=}& \mathbf{G}\left(\mathbf{v}\right)\\ & \mathbf{=}& \frac{\mathbf{1}}{\mathbf{2}}\left(\mathbf{-}{\mathbf{v}}^{\mathbf{2}}\mathbf{+}\mathbf{v}\right)\end{array}$

So, the given equation is homogeneous.

Now, rewriting the equation,

$\begin{array}{rcl}\frac{\mathbf{dy}}{\mathbf{d}胃}& \mathbf{=}& \mathbf{-}\frac{{\mathbf{y}}^{\mathbf{2}}}{\mathbf{2}{胃}^{\mathbf{2}}}\mathbf{+}\frac{\mathbf{y}}{\mathbf{2}胃}\\ \frac{\mathbf{dy}}{\mathbf{d}胃}\mathbf{+}\left(\mathbf{-}\frac{\mathbf{1}}{\mathbf{2}胃}\right)\mathbf{y}& \mathbf{=}& \left(\mathbf{-}\frac{\mathbf{1}}{\mathbf{2}{胃}^{\mathbf{2}}}\right){\mathbf{y}}^{\mathbf{2}}\\ \frac{\mathbf{dy}}{\mathbf{d}胃}\mathbf{+}\mathbf{P}\left(胃\right)\mathbf{y}& \mathbf{=}& \mathbf{Q}\left(胃\right){\mathbf{y}}^{\mathbf{n}}\\ & & \end{array}$

It seems that the given equation is Bernoulli.

Therefore, the given equation is the form of homogenous and Bernoulli equation.