solve (x^3*y^3+x^2*y^2+xy+1)ydx + (x^3*y^3-x^2*y^2-xy+1)xdy=0

Question

solve (x^3*y^3+x^2*y^2+xy+1)ydx + (x^3*y^3-x^2*y^2-xy+1)xdy=0

Answer 1

∂/∂y(x³y⁴ + +x²y³ + xy² + y) = 4x³y³ + 3x²y² + 2xy + 1

∂/∂x(x⁴y³ + +x³y² + x²y + x) = 4x³y³ + 3x²y² + 2xy + 1

So, we have exact equation.

F(x,y) = ∫(x³y⁴ + x²y³ + xy² + y)dx + φ(y) = x⁴y⁴/4 + x³y³/3 + x²y²/2 + xy + φ(y);

∂F/∂y = x⁴y³ + x³y² + x²y + x + φ'(y) = x⁴y³ + +x³y² + x²y + x; φ'(y) = 0; φ(y) = C(const)

F(x,y) = x⁴y⁴/4 + x³y³/3 + x²y²/2 + xy + C