Question

In problem$\mathbf{1}\mathbf{-}\mathbf{6}$ , determine whether the differential equation is separable$\frac{\mathbf{d}\mathbf{y}}{\mathbf{d}\mathbf{x}}\mathbf{=}\frac{\mathbf{y}\mathbf{}{\mathbf{e}}^{\mathbf{x}\mathbf{+}\mathbf{y}}}{{\mathbf{x}}^{\mathbf{2}}\mathbf{+}\mathbf{2}}$.

Short answer

The differential equation$\frac{dy}{dx}=\frac{y{e}^{x+y}}{x^2+y}$ is separable.

Step-by-step solution

Concept of Separable Differential Equation

A first-order ordinary differential equation $\frac{\mathbf{d}\mathbf{y}}{\mathbf{d}\mathbf{x}}\mathbf{=}\mathit{f}\left(x,y\right)$is referred to as separable if the function in the right-hand side of the equation is expressed as a product of two functions role="math" localid="1654771361248" $\mathit{g}\mathbf{\left(}\mathbf{x}\mathbf{\right)}$that is a function of x alone and h(y) that is a function of alone.

Mathematically, the equation is separable, when f(x,y) = g(x) + h(y).

Determining whether the equation is Separable or not

The given equation is

$\frac{dy}{dx}=\frac{y{e}^{x+y}}{x^2+y}...........\left(1\right)$

The function in the right-hand side of equation (1) is

$$\begin{gathered} f\left(x,y\right)=\frac{y{e}^{x+y}}{x^2+2} \\ =\frac{y{e}^x{e}^y}{x^2+2} \\ =\frac{e^x}{x^2+2}y{e}^y \end{gathered}$$

This function can be written as a product of two functions and defined as,

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

Therefore, the given differential equation is separable.