4.17.33 $y^{\prime }(x{)}^2-3xy(x{)}^{2/3}{y}^{\prime }(x)+9y(x{)}^{5/3}=0$

ODE
$$y^{\prime }(x{)}^2-3xy(x{)}^{2/3}{y}^{\prime }(x)+9y(x{)}^{5/3}=0$$ ODE Classification

[[_1st_order, _with_linear_symmetries]]

Book solution method
Change of variable

Mathematica ✓
cpu = 0.971818 (sec), leaf count = 175

$$\{\text{Solve}[\frac{1}{6}(-6c_1+\frac{{(x^2-4\sqrt[3]{y(x)})}^{3/2}y(x{)}^2(6\log (\sqrt{x^2-4\sqrt[3]{y(x)}}+x)-\log (y(x)))}{{((x^2-4\sqrt[3]{y(x)})y(x{)}^{4/3})}^{3/2}}+\log (y(x)))=0,y(x)],\text{Solve}[\frac{1}{6}(-6c_1+\frac{{(x^2-4\sqrt[3]{y(x)})}^{3/2}y(x{)}^2(\log (y(x))-6\log (\sqrt{x^2-4\sqrt[3]{y(x)}}+x))}{{((x^2-4\sqrt[3]{y(x)})y(x{)}^{4/3})}^{3/2}}+\log (y(x)))=0,y(x)]\}$$

Maple ✓
cpu = 2.484 (sec), leaf count = 142

$$\{\sqrt[3]{y\left(x\right)}-\frac{x^2}{4}=0,\ln \left(x\right)-\frac{1}{6}\ln (4\sqrt[3]{\frac{y\left(x\right)}{x^6}}-1)+\frac{1}{6}\ln \left(\frac{y\left(x\right)}{x^6}\right)-1\sqrt{-4{\left(\frac{y\left(x\right)}{x^6}\right)}^{5/3}+{\left(\frac{y\left(x\right)}{x^6}\right)}^{\frac{4}{3}}}\mathit{A}\mathit{r}\mathit{t}\mathit{a}\mathit{n}\mathit{h}\left(\sqrt{-4\sqrt[3]{\frac{y\left(x\right)}{x^6}}+1}\right){\left(\frac{y\left(x\right)}{x^6}\right)}^{-\frac{2}{3}}\frac{1}{\sqrt{-4\sqrt[3]{\frac{y\left(x\right)}{x^6}}+1}}-\frac{1}{6}\ln (16{\left(\frac{y\left(x\right)}{x^6}\right)}^{2/3}+4\sqrt[3]{\frac{y\left(x\right)}{x^6}}+1)+\frac{1}{6}\ln (64\frac{y\left(x\right)}{x^6}-1)-\mathit{\_}\mathit{C}\mathit{1}=0\}$$ Mathematica raw input

DSolve[9*y[x]^(5/3) - 3*x*y[x]^(2/3)*y'[x] + y'[x]^2 == 0,y[x],x]

Mathematica raw output

{Solve[(-6*C[1] + Log[y[x]] + ((6*Log[x + Sqrt[x^2 - 4*y[x]^(1/3)]] - Log[y[x]])
*(x^2 - 4*y[x]^(1/3))^(3/2)*y[x]^2)/((x^2 - 4*y[x]^(1/3))*y[x]^(4/3))^(3/2))/6 =
= 0, y[x]], Solve[(-6*C[1] + Log[y[x]] + ((-6*Log[x + Sqrt[x^2 - 4*y[x]^(1/3)]] 
+ Log[y[x]])*(x^2 - 4*y[x]^(1/3))^(3/2)*y[x]^2)/((x^2 - 4*y[x]^(1/3))*y[x]^(4/3)
)^(3/2))/6 == 0, y[x]]}

Maple raw input

dsolve(diff(y(x),x)^2-3*x*y(x)^(2/3)*diff(y(x),x)+9*y(x)^(5/3) = 0, y(x),'implicit')

Maple raw output

y(x)^(1/3)-1/4*x^2 = 0, ln(x)-1/6*ln(4*(y(x)/x^6)^(1/3)-1)+1/6*ln(y(x)/x^6)-(-4*
(y(x)/x^6)^(5/3)+(y(x)/x^6)^(4/3))^(1/2)/(y(x)/x^6)^(2/3)/(-4*(y(x)/x^6)^(1/3)+1
)^(1/2)*arctanh((-4*(y(x)/x^6)^(1/3)+1)^(1/2))-1/6*ln(16*(y(x)/x^6)^(2/3)+4*(y(x
)/x^6)^(1/3)+1)+1/6*ln(64*y(x)/x^6-1)-_C1 = 0