Question

(2y^2-4x+5)dx+(4-2y+4xy)dy=0
For exactness, and solve it if it is exact.

Answer 1

Given

$$(2y^2-4x+5)dx+(4-2y+4xy)dy=0$$

Let $M(x,y)=2y^2-4x+5$ and $N(x,y)=4-2y+4xy$

Thus, differential equation becomes

$$Mdx+Ndy=0$$

Now, to check exactness we have to check

$$\partial_yM=\partial_xN$$

or not.

In this case ,

$$\partial_yM=4y=\partial_xN$$

Thus,

$$(2y^2-4x+5)dx+(4-2y+4xy)dy=0$$

is exact differential equation.

Thus,

$$(2y^2-4x+5)dx+(4-2y+4xy)dy=0\\ \implies (-4x+5)dx+(4-2y)dy+d(2xy^2)=0\\ \implies \int(-4x+5)dx+\int(4-2y)dy+\int d(2xy^2)=C\\ \implies -2x^2+5x+4y-y^2+2xy^2=C$$