I've been on this question for two days and I can't seem to get past the last integral. The natural logarithm of y seems to be a problem $\mathrm{ y\frac{d^2y}{dx^2} + \frac{dy}{dx} + \Bigl(\frac{dy}{dx}\Bigl)^3} = 0\\ \mathrm {substiting \, dy/dx = p, \, d²y/dx² = pdp/dy}\\ \mathrm{ reduces \, the \, equation\, to} \\ \mathrm{ yp\frac{dp}{dy} + p +p^3= 0} \implies \mathrm{y\frac{dp}{dy} + 1 +p^2= 0}\\ \mathrm{\frac{dp}{1+p^2} + \frac{dy}{y}}= 0 \implies \mathrm{ tan^{-1}p + Iny = C}\\ \mathrm{p = tan{(c - Iny) }} \implies\mathrm{\frac{dy}{tan(c-Iny)} = dx}$\.
Is there another method to go about it?