A differential equation is exact if it is of the form
partial/{partialx}[F(x,y)]dx+partial/{partialy}[F(x,y)]dy=0
If the differential equation is exact, then we would have
partial/{partialx}F{x,y}=5x+4y and partial/{partialy}F{x,y}=4x-8y^3
The order of partial derivatives on F(x,y) shouldn't matter so we must check whether the above conditions satisfy the equation below
partial/{partialy}[partial/{partialx}F{x,y}]=partial/{partialx}[partial/{partialy}F{x,y}]
Substitute,
partial/{partialy}[5x+4y]quad?=?\quadpartial/{partialx}[4x-8y^3]
4 \quad ?=?\quad4
Both sides are equal. Therefore there is a function, F(x,y), that can meet the requirements that
partial/{partialx}F{x,y}=5x+4y and partial/{partialy}F{x,y}=4x-8y^3
and the differential equation is exact.
partial/{partialx}F{x,y}=5x+4y
\impliesF(x,y)=5/2x^2+4xy+G(y)
where G(y) is a function of y only. When integrating, we pick up an arbitrary constant. Since this is a partial derivative with respect to x, y is considered constant and the arbitrary constant can contain terms with y.
partial/{partialy}F{x,y}=4x-8y^3
\implies F(x,y)=4xy-2y^4+Z(x)
where Z(x) is a function that only depends on x
Comparing the two, we can set G(y)=-2y^4+C and Z(x)=5/2x^2+C
where C is a constant, and then the two expressions for F(x,y) would agree with each other. The solution to the exact differential equation is then.
F(x,y)=5/2x^2+4xy-2y^4+C