##### 4.17.33 $$y'(x)^2-3 x y(x)^{2/3} y'(x)+9 y(x)^{5/3}=0$$

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

[[_1st_order, _with_linear_symmetries]]

Book solution method
Change of variable

Mathematica
cpu = 2.06697 (sec), leaf count = 165

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

Maple
cpu = 4.954 (sec), leaf count = 266

$\left [y \left (x \right ) = \frac {x^{6}}{64}, \ln \left (x \right )+\frac {\ln \left (\frac {64 y \left (x \right )}{x^{6}}-1\right )}{6}-\frac {\ln \left (4 \left (\frac {y \left (x \right )}{x^{6}}\right )^{\frac {1}{3}}-1\right )}{6}-\frac {\ln \left (16 \left (\frac {y \left (x \right )}{x^{6}}\right )^{\frac {2}{3}}+4 \left (\frac {y \left (x \right )}{x^{6}}\right )^{\frac {1}{3}}+1\right )}{6}+\frac {\ln \left (\frac {y \left (x \right )}{x^{6}}\right )}{6}+\frac {\sqrt {-4 \left (\frac {y \left (x \right )}{x^{6}}\right )^{\frac {5}{3}}+\left (\frac {y \left (x \right )}{x^{6}}\right )^{\frac {4}{3}}}\, \arctanh \left (\sqrt {-4 \left (\frac {y \left (x \right )}{x^{6}}\right )^{\frac {1}{3}}+1}\right )}{\left (\frac {y \left (x \right )}{x^{6}}\right )^{\frac {2}{3}} \sqrt {-4 \left (\frac {y \left (x \right )}{x^{6}}\right )^{\frac {1}{3}}+1}}-\textit {\_C1} = 0, \ln \left (x \right )+\frac {\ln \left (\frac {64 y \left (x \right )}{x^{6}}-1\right )}{6}-\frac {\ln \left (4 \left (\frac {y \left (x \right )}{x^{6}}\right )^{\frac {1}{3}}-1\right )}{6}-\frac {\ln \left (16 \left (\frac {y \left (x \right )}{x^{6}}\right )^{\frac {2}{3}}+4 \left (\frac {y \left (x \right )}{x^{6}}\right )^{\frac {1}{3}}+1\right )}{6}+\frac {\ln \left (\frac {y \left (x \right )}{x^{6}}\right )}{6}-\frac {\sqrt {-4 \left (\frac {y \left (x \right )}{x^{6}}\right )^{\frac {5}{3}}+\left (\frac {y \left (x \right )}{x^{6}}\right )^{\frac {4}{3}}}\, \arctanh \left (\sqrt {-4 \left (\frac {y \left (x \right )}{x^{6}}\right )^{\frac {1}{3}}+1}\right )}{\left (\frac {y \left (x \right )}{x^{6}}\right )^{\frac {2}{3}} \sqrt {-4 \left (\frac {y \left (x \right )}{x^{6}}\right )^{\frac {1}{3}}+1}}-\textit {\_C1} = 0\right ]$ 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[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*C[1], y[x]
], Solve[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*C[1], 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))

Maple raw output

[y(x) = 1/64*x^6, ln(x)+1/6*ln(64*y(x)/x^6-1)-1/6*ln(4*(y(x)/x^6)^(1/3)-1)-1/6*l
n(16*(y(x)/x^6)^(2/3)+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))-_C1 = 0, ln(x)+1/6*ln(64*y(x)/x^6-1)-1/6*ln(4*(y
(x)/x^6)^(1/3)-1)-1/6*ln(16*(y(x)/x^6)^(2/3)+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))-_C1 = 0]