Question

Classify each differential equation as separable, exact, linear, homogeneous, or Bernoulli. Some equations may be more than one kind. Do not solve.

$\frac{\mathbf{y}}{{\mathbf{x}}^{\mathbf{2}}}\frac{\mathbf{dy}}{\mathbf{dx}}\mathbf{+}{\mathbf{e}}^{\mathbf{2}{\mathbf{x}}^{\mathbf{3}}\mathbf{+}{\mathbf{y}}^{\mathbf{2}}}\mathbf{=}\mathbf{0}$

Short answer

The equation is separable and has no additional forms.

$\begin{array}{l}\mathrm{g}\left(\mathrm{x}\right)=-{\mathrm{x}}^2{\mathrm{e}}^{{\mathrm{x}}^2}\\ \mathrm{h}\left(\mathrm{y}\right)=\frac{{\mathrm{e}}^{{\mathrm{y}}^2}}{\mathrm{y}}\end{array}$

Step-by-step solution

Define Bernoulli equation

A Bernoulli equation looks like this$\frac{\mathbf{d}\mathbf{y}}{\mathbf{d}\mathbf{x}}\mathbf{}\mathbf{+}\mathbf{}\mathbf{P}\mathbf{\left(}\mathbf{x}\mathbf{\right)}\mathbf{y}\mathbf{}\mathbf{=}\mathbf{}\mathbf{f}\mathbf{\left(}\mathbf{x}\mathbf{\right)}{\mathbf{y}}^{\mathbf{n}}$

The equation can be solved as a First Order Linear Differential Equation when n = 0

Find the equation if they are separable or not.

Show the equation in the form of$\frac{dy}{dx}=g\left(x\right)h\left(y\right)$

$\frac{\mathrm{y}}{{\mathrm{x}}^2}\frac{\mathrm{dy}}{\mathrm{dx}}+{\mathrm{e}}^{2{\mathrm{x}}^2+{\mathrm{y}}^2}=0$

$\frac{y}{x^2}\frac{dy}{dx}=-e^{2x^2+y^2}$

$\frac{\mathrm{dy}}{\mathrm{dx}}=-\frac{{\mathrm{x}}^2{\mathrm{e}}^{2{\mathrm{x}}^2+{\mathrm{y}}^2}}{\mathrm{y}}$

$=-\frac{x^2{e}^{2x^2}\cdot e^{y^2}}{y}$

Which is in the form oflocalid="1668419880772" $\frac{\mathrm{dy}}{\mathrm{dx}}=\mathrm{g}\left(\mathrm{x}\right)\mathrm{h}\left(\mathrm{y}\right)$

$\begin{array}{c}g\left(x\right)=-x^2{e}^{x^2}\\ h\left(y\right)=\frac{e^{y^2}}{y}\end{array}$

Hence the equation is separable and has no additional forms.