Question

In Problems determine whether the given differential equation is exact. If it is exact, solve it.$\mathbf{(}\mathbf{2}\mathit{x}\mathbf{+}\mathit{y}\mathbf{)}\mathit{d}\mathit{x}\mathbf{-}\mathbf{(}\mathit{x}\mathbf{+}\mathbf{6}\mathit{y}\mathbf{)}\mathit{d}\mathit{y}\mathbf{=}\mathbf{0}$

Short answer

The given differential equation is not exact

Step-by-step solution

Given Information.

The given equation is$(2x+y)dx-(x+6y)dy=0$. An equation of the form$Mdx+Ndy=f\left(\text{x}\right)$is exact only if it satisfieslocalid="1663834829521" $\frac{\partial \text{M}}{\partial \text{y}}=\frac{\partial \text{N}}{\partial \text{x}}$

Determining the exactness of the differential equation

Find M and N using given equation

$\begin{array}{l}M(x,y)=\left(2x+y\right)\\ N(x,y)=-(x+6y)\end{array}$

Findlocalid="1663834844278" $\frac{\partial \text{M}}{\partial \text{y}}$and localid="1663834849983" $\frac{\partial \text{N}}{\partial \text{x}}$.

$\begin{array}{l}\frac{\partial M(x,y)}{\partial y}=1\\ \frac{\partial N(x,y)}{\partial x}=-1\end{array}$

As, $\frac{\partial M(x,y)}{\partial y}\ne \frac{\partial N(x,y)}{\partial x}$so, the equation is not exact