Question

In Problems 1-20 determine whether the given differential equation is exact. If it is exact, solve it.

$\left(ylny-e^{-xy}\right)\mathit{d}\mathit{x}\mathbf{+}\left(\frac{1}{y}+xlny\right)\mathit{d}\mathit{y}\mathbf{=}\mathbf{0}$

Short answer

The given equation is not exact.

Step-by-step solution

Given information

The given equation is, $\left(ylny-e^{-xy}\right)dx+\left(\frac{1}{y}+xlny\right)dy=0$.

Compare the given equation with $Mdx+Ndy=0$

$$\begin{gathered} M(x,y)=ylny-e^{-xy} \\ N(x,y)=\frac{1}{y}+x\ln y \end{gathered}$$

Condition for exactness

An equation of the form$Mdx+Ndy=0$ , is said to be exact, if$\frac{\partial M}{\partial y}=\frac{\partial N}{\partial x}$ .

Determining the exactness of the differential equation

Checking whether the differential equation is exact by finding $\frac{\partial M}{\partial y},\frac{\partial N}{\partial x}$

$$\begin{gathered} \frac{\partial M}{\partial y}=1+lny+x{e}^{-xy} \\ \frac{\partial N}{\partial x}=\ln y \end{gathered}$$

It can be observed that $\frac{\partial M}{\partial y}\ne \frac{\partial N}{\partial x}$.

Therefore, the equation is not exact.