Question

(1 point) In this problem we consider an equation in differential form M dx + N dy = 0. (8x + 8y)dx + (8x + 4y)dy = 0 Find My = 8 Nx = 8 If the problem is exact find a function F(x, y) whose differential, dF(x, y) is the left hand side of the differential equation. That is, level curves F(x, y) = C, give implicit general solutions to the differential equation. If the equation is not exact, enter NE otherwise find F(x, y) (note you are not asked to enter C) F(x, y) =

Answer 1

The ODE is exact because $M_y=N_x$. Then

$F_x=8x+8y\implies F(x,y)=4x^2+8xy+g(y)$

$F_y=8x+g'(y)=8x+4y\implies g'(y)=4y\implies g(y)=2y^2+C$

So we have

$F(x,y)=4x^2+8xy+2y^2=C$