Chapter 16, Problem 1RE

To determine

Whether the differential equation $\left(y+x^3+x{y}^2\right)dx-xdy=0$ is exact or not.

Answer to Problem 1RE

Solution:

The differential equation $\left(y+x^3+x{y}^2\right)dx-xdy=0$ isnot exact.

Explanation of Solution

Given information:

The given differential equation is $\left(y+x^3+x{y}^2\right)dx-xdy=0$.

Concept Used:

Let M and N have continuous partial derivatives on an open disk R. The differential equation

$M(x,y)dx+N(x,y)dy=0$

is exact if and only if

$\frac{\partial M}{\partial y}=\frac{\partial N}{\partial x}$

Calculation:

First we will compare differential equation $\left(y+x^3+x{y}^2\right)dx-xdy=0$ with the $M(x,y)dx+N(x,y)dy=0$.

We have

$\begin{array}{l}M=y+x^3+x{y}^2\\ N=-x\\ \frac{\partial M}{\partial y}=\frac{\partial (y+x^3+x{y}^2)}{\partial y}\\ \frac{\partial M}{\partial y}=1+2y\\ \frac{\partial N}{\partial x}=\frac{\partial (-x)}{\partial x}\\ \frac{\partial N}{\partial x}=-1\end{array}$

Since $\frac{\partial M}{\partial y}\ne \frac{\partial N}{\partial x}$, hence this differential equation is not exact.