P(x,y)=5x^4+3xy^2-y^3
Q(x,y)=3x^2y-3xy^2+y^2
∂P/∂y=6xy-3y^2=∂Q/∂x
∂P/∂y=∂Q/∂x
设通解为R(x,y)=C
∂R/∂x=P(x,y)
R(x,y)=∫P(x,y)dx
=∫(5x^4+3xy^2-y^3)dx
=x^5+3/2x^2y^2-xy^3+S(y)
∂R/∂y=3x^2y-3xy^2+S'(y)
=Q(x,y)
=3x^2y-3xy^2+y^2
S'(y)=y^2
S(y)=y^3/3+C
通解为
x^5+3/2x^2y^2-xy^3+y^3/3=C