Question

1/(xy²+y⁴) is an integrating factor for the differential equation (x²y+y²)dx+(y³-x³)dy=0. Is it true or false. Give reasons for your answer

Answer 1

$\displaystyle \textsf{It is not an integrating factor}\\ (x^2 y + y^2)\mathrm{d}x + (y^3 - x^3) \mathrm{d}y = 0\\ \frac{\mathrm{d}y}{\mathrm{d}x} + \frac{(x^2 + y)y}{y^3 - x^3} = 0\\ \frac{\mathrm{d}}{\mathrm{d}x}\left(\frac{y}{xy^2 + y^4}\right) \neq \frac{\mathrm{d}y}{\mathrm{d}x} + \frac{(x^2 + y)y}{y^3 - x^3} \\ \textsf{Or}\\ (x^2 y + y^2)\mathrm{d}x + (y^3 - x^3) \mathrm{d}y = 0\\ y(x^2\mathrm{d}x + y^2\mathrm{d}y) + y^2\mathrm{d}x - x^3\mathrm{d}y = 0\\ \textsf{Multiply both sides by}\,\, \frac{1}{xy^2 + y^4} \\ \frac{x\mathrm{d}x + y^2\mathrm{d}y}{y(x + y^2)} + \frac{y^2\mathrm{d}x - x^3\mathrm{d}y}{y^2(x + y^2)} = 0\\ \frac{\mathrm{d}\left(\frac{x^2}{2} + \frac{y^3}{3}\right)}{y(x + y^2)} + \frac{y^2\mathrm{d}x - x^3\mathrm{d}y}{y^2(x + y^2)} = 0\\ \textsf{This cannot be simplified further}\\ \textsf{to obtain a solution of the ODE}\\ \textsf{Therefore}\,\, \frac{1}{xy^2 + y^4}\,\,\textsf{is not an integrating factor}$