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 = 0.971818 (sec), leaf count = 175

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

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

\[ \left \{ \sqrt [3]{y \relax (x ) }-{\frac {{x}^{2}}{4}}=0,\ln \relax (x ) -{\frac {1}{6}\ln \left (4\,\sqrt [3]{{\frac {y \relax (x ) }{{x}^{6}}}}-1 \right ) }+{\frac {1}{6}\ln \left ({\frac {y \relax (x ) }{{x}^{6}}} \right ) }-{1\sqrt {-4\, \left ({\frac {y \relax (x ) }{{x}^{6}}} \right ) ^{5/3}+ \left ({\frac {y \relax (x ) }{{x}^{6}}} \right ) ^{{\frac {4}{3}}}}{\it Artanh} \left (\sqrt {-4\,\sqrt [3]{{\frac {y \relax (x ) }{{x}^{6}}}}+1} \right ) \left ({\frac {y \relax (x ) }{{x}^{6}}} \right ) ^{-{\frac {2}{3}}}{\frac {1}{\sqrt {-4\,\sqrt [3]{{\frac {y \relax (x ) }{{x}^{6}}}}+1}}}}-{\frac {1}{6}\ln \left (16\, \left ({\frac {y \relax (x ) }{{x}^{6}}} \right ) ^{2/3}+4\,\sqrt [3]{{\frac {y \relax (x ) }{{x}^{6}}}}+1 \right ) }+{\frac {1}{6}\ln \left (64\,{\frac {y \relax (x ) }{{x}^{6}}}-1 \right ) }-{\it \_C1}=0 \right \} \] 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[(-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