Answer to Question #105778 in Differential Equations for Simon

Question

Show that the following equation is exact
( x² + y²) dx + 2xydy = 0

Answer 1

We known that, The necessary and sufficient condition for the differential equation

$Mdx+Ndy=0$

where $M$ and $N$ are function of $x$ and $y$, to be exact is that

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

According to the question,

$M=x^2+y^2$ and $N=2xy$

Therefore,$\frac {\partial M}{\partial y}=2y$ and $\frac {\partial N}{\partial x}=2y$ .

Hence, the given equation $(x^2+y^2)dx+2xydy=0$

is exact.