Equation (3y^2 - x)dx + (2y^3 - 6xy)dy = 0 has an integrating factor in the form M(x,y) = (x + y). (a) Show that this equation is not exact(complete). (b) Solve the equation by using M.

Question

Equation (3y^2 - x)dx + (2y^3 - 6xy)dy = 0 has an integrating factor in the form M(x,y) = (x + y).

(a) Show that this equation is not exact(complete).

(b) Solve the equation by using M.

Answer 1

∂/∂y(3y² - x) = 6y ≠ - 6y = ∂/∂x(2y³ - 6xy). So, this is not exact equation. Now, let multiply by M(x,y) = (x + y);

(x + y)(3y² - x)dx + (x + y)(2y³ - 6xy)dy = 0.

Now, ∂/∂y[(x + y)(3y² - x)] = 3y² - x + 6yx + 6y² = 9y² - x + 6xy

∂/∂x[(x + y)(2y³ - 6xy)] = 2y³ - 6xy +