# Non Linear ODE and change of variable

$$\displaystyle y''=x^{-\frac{1}{2}}y^{\frac{3}{2}}$$ using $$\displaystyle t=y(x)^{\frac{1}{6}}\frac{\sqrt{x}}{144^{\frac{1}{6}}}$$ and $$\displaystyle u(t)=-\left(\frac{16}{3}\right)^{\frac{1}{3}}\frac{y'(x)}{y(x)^{\frac{3}{4}}}$$