Solve by inspection...

Question

1. ( x³+ x y² +y ) d x + ( y³ + x² y + x ) d y = 0
2. x⁴ y' = -x³ y - csc(xy)

Answer 1

(1) is exact because Mdx + Ndy = 0 has dM/dy = dN/dx There is a solution of form Ψ(x,y) = C

Integrate Mdx and get Ψ = x⁴/4 + x²y²/2 +xy + c(y). Take the derivative wrt y and you get

x²y +x +c'(y) = N or c'(y) = y³ so c(y) = y⁴/4 + C

Ψ(x,y) = x⁴/4 + x²y²/2 +xy + y⁴/4 = C

2) I cannot solve this by inspection. I used symbolab for the solution. It was not obvious.