Question

Solve 2(2y^2+yz -z^2)dx+x(4y +z)dy+x(y -2z)dz= 0

Answer 1

$2(2y^2+yz -z^2)dx+x(4y +z)dy+x(y -2z)dz= 0$

condition if the equation is integrable:

$P(\partial R/\partial y-\partial Q/\partial z)-Q(\partial R/\partial x-\partial P/\partial z)+R(\partial Q/\partial x-\partial P/\partial y)=0$

we have:

$P=2(2y^2+yz -z^2),Q=x(4y +z),R=x(y -2z)$

$\partial R/\partial y=x,\partial Q/\partial z=x,\partial R/\partial x=y$

$\partial P/\partial z=2y-4z,\partial Q/\partial x=4y+z,\partial P/\partial y=8y+2z$

$-x(4y +z)(-y+4z)+x(y -2z)(-4y-z)=-2xz(4y+z)\neq 0$

so, the equation is not integrable