Question

In Problems 31–34 verify that the indicated expression is an implicit

solution of the given differential equation.

$\mathit{y}\mathbf{\text{'}}\mathbf{(}\mathbf{1}\mathbf{-}\mathit{x}\mathit{y}\mathbf{)}\mathbf{=}{\mathit{y}}^{\mathbf{2}}\mathbf{;}\mathit{y}\mathbf{=}{\mathit{e}}^{\mathbf{x}\mathbf{y}}$

Short answer

The indicated expression is an implicit solution for the given differential equation.

Step-by-step solution

Definition of differential equation

A differential equation is defined as an equation that consists of the derivatives of one or more dependent functions with respect to one or more independent functions.

Taking derivative with respect to x

Given an implicit solution, $y=e^{xy}$.

Take the logarithm on both sides first, then the derivative of the expression with respect to x, and simplify in the end to get the desired differential equation.

Taking logarithm on both sides,

$\ln y=xy$

Taking the derivative with respect to x,

$\frac{1}{y}y\text{'}=xy\text{'}+y$

Taking $y\text{'}$as a common factor,

$y\text{'}(\frac{1}{y}-x)=y$

Simplify, we get

$y\text{'}(1-xy)=y^2$

Hence, verified.

Therefore, the indicated expression is an implicit solution for the given differential equation.