Internal problem ID [7978]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, linear first order
Problem number: 398.
ODE order: 1.
ODE degree: 2.
CAS Maple gives this as type [[_1st_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {\left (y^{\prime }\right )^{2}-3 x y^{\frac {2}{3}} y^{\prime }+9 y^{\frac {5}{3}}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.481 (sec). Leaf size: 141
dsolve(diff(y(x),x)^2-3*x*y(x)^(2/3)*diff(y(x),x)+9*y(x)^(5/3) = 0,y(x), singsol=all)
\begin{align*} y \relax (x ) = \frac {x^{6}}{64} \\ y \relax (x ) = 0 \\ \ln \relax (x )+\frac {\ln \left (\frac {64 y \relax (x )}{x^{6}}-1\right )}{6}-\frac {\ln \left (16 \left (\frac {y \relax (x )}{x^{6}}\right )^{\frac {2}{3}}+4 \left (\frac {y \relax (x )}{x^{6}}\right )^{\frac {1}{3}}+1\right )}{6}-\frac {\ln \left (4 \left (\frac {y \relax (x )}{x^{6}}\right )^{\frac {1}{3}}-1\right )}{6}+\frac {\ln \left (\frac {y \relax (x )}{x^{6}}\right )}{6}+\frac {\sqrt {-4 \left (\frac {y \relax (x )}{x^{6}}\right )^{\frac {5}{3}}+\left (\frac {y \relax (x )}{x^{6}}\right )^{\frac {4}{3}}}\, \arctanh \left (\sqrt {-4 \left (\frac {y \relax (x )}{x^{6}}\right )^{\frac {1}{3}}+1}\right )}{\left (\frac {y \relax (x )}{x^{6}}\right )^{\frac {2}{3}} \sqrt {-4 \left (\frac {y \relax (x )}{x^{6}}\right )^{\frac {1}{3}}+1}}-c_{1} = 0 \\ \end{align*}
✓ Solution by Mathematica
Time used: 1.588 (sec). Leaf size: 165
DSolve[9*y[x]^(5/3) - 3*x*y[x]^(2/3)*y'[x] + y'[x]^2==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} \text {Solve}\left [\frac {1}{6} \log (y(x))-\frac {\sqrt {\left (x^2-4 \sqrt [3]{y(x)}\right ) y(x)^{4/3}} \tanh ^{-1}\left (\frac {x}{\sqrt {x^2-4 \sqrt [3]{y(x)}}}\right )}{\sqrt {x^2-4 \sqrt [3]{y(x)}} y(x)^{2/3}}=c_1,y(x)\right ] \\ \text {Solve}\left [\frac {\sqrt {\left (x^2-4 \sqrt [3]{y(x)}\right ) y(x)^{4/3}} \tanh ^{-1}\left (\frac {x}{\sqrt {x^2-4 \sqrt [3]{y(x)}}}\right )}{\sqrt {x^2-4 \sqrt [3]{y(x)}} y(x)^{2/3}}+\frac {1}{6} \log (y(x))=c_1,y(x)\right ] \\ y(x)\to 0 \\ \end{align*}