Question

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

solution of the given differential equation.

$\mathit{x}\frac{\mathbf{d}\mathbf{y}}{\mathbf{d}\mathbf{x}}\mathbf{+}\mathit{y}\mathbf{=}\frac{\mathbf{1}}{{\mathbf{y}}^{\mathbf{2}}}\mathbf{;}{\mathit{x}}^{\mathbf{3}}{\mathit{y}}^{\mathbf{3}}\mathbf{=}{\mathit{x}}^{\mathbf{3}}\mathbf{+}\mathbf{1}$

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 the equation consisting of the derivatives of one or more than one dependent function with respect to one or more than one independent function.

Taking derivative with respect to x

Given an implicit solution,$x^3{y}^3=x^3+1$

Differentiate the expression with respect to x on both sides and rewrite it to the given differential equation.

Taking the derivative with respect to x.

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

Divide both sides by$3x^2{y}^2$ to get back the differential equation,

$x\frac{dy}{dx}+y=\frac{1}{y^2}$

Hence, verified.

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