Question

Determine whether the given differential equation is exact. If it is exact, solve it.

$\left(x^2-y^2\right)\mathit{d}\mathit{x}\mathbf{+}\mathbf{(}{\mathit{x}}^{\mathbf{2}}\mathbf{-}\mathbf{2}\mathit{x}\mathit{y}\mathbf{)}\mathit{d}\mathit{y}\mathbf{=}\mathbf{0}$

Short answer

The given differential equation is not exact

Step-by-step solution

Define exact differential equation

The given equation is$(x^2-y^2)dx+\left(x^2-2xy\right)dy=0$. An equation of the form $Mdx+Ndy=f\left(\text{x}\right)$is exact only if it satisfieslocalid="1663835509821" $\frac{\partial \text{M}}{\partial \text{y}}=\frac{\partial \text{N}}{\partial \text{x}}$

Determining the exactness of the differential equation

Find M and N using the given equation

$\begin{array}{l}M(x,y)=x^2-y^2\\ N(x,y)=x^2-2xy\end{array}$

Find role="math" localid="1663829682295" $\frac{\partial M}{\partial y}$and$\frac{\partial N}{\partial x}$

$\begin{array}{l}\frac{\partial M}{\partial y}=-2y\\ \frac{\partial N}{\partial x}=2x-2y\end{array}$

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