$2+(2)=0$

$2$

y

$+$

$($

$2$

x

$$

y

$)$

d

y

d

x

$=$

$0$

$,$

which is in the general form $(,)+(,)=0$

P

$($

x

$,$

y

$)$

$+$

Q

$($

x

$,$

y

$)$

d

y

d

x

$=$

$0$

$.$ We can find the two following expressions for $(,)$

$\backslash$psi

$($

x

$,$

y

$)$

:

$(,)=(,)=2=2+()$

$\backslash$psi

$($

x

$,$

y

$)$

$=$

$$

P

$($

x

$,$

y

$)$

d

x

$=$

$$

$2$

y

d

x

$=$

$2$

x

y

$+$

f

$($

y

$)$

$.$

$(,)=(,)=(2)=2122+()$

$\backslash$psi

$($

x

$,$

y

$)$

$=$

$$

Q

$($

x

$,$

y

$)$

d

y

$=$

$$

$($

$2$

x

$$

y

$)$

d

y

$=$

$2$

x

y

$$

$1$

$2$

y

$2$

$+$

g

$($

x

$)$

$.$

By matching the above two expressions for $(,)$

$\backslash$psi

$($

x

$,$

y

$)$

$,$ give $()$

f

$($

y

$)$

$,()$

g

$($

x

$)$

and the general solution of the exact differential equation.

$()=$

f

$($

y

$)$

$=$

$()=$

g

$($

x

$)$

$=$

General solution $(,)=$

$\backslash$psi

$($

x

$,$

y

$)$

$=$

$=$

$=$

c

$.$