Internal problem ID [3560]
Book: Ordinary differential equations and their solutions. By George Moseley Murphy.
1960
Section: Various 29
Problem number: 829.
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: 2.781 (sec). Leaf size: 142
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 (4 \left (\frac {y \relax (x )}{x^{6}}\right )^{\frac {1}{3}}-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 (\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: 17.079 (sec). Leaf size: 701
DSolve[(y'[x])^2-3 x y[x]^(2/3) y'[x]+9 y[x]^(5/3)==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} \text {Solve}\left [\frac {8 x^2 \log (y(x))-6 \sqrt {x^4} \log \left (x^2 \sqrt {x^2-4 \sqrt [3]{y(x)}}\right )-3 \sqrt {x^4} \log \left (4 \sqrt [3]{y(x)}-x^2\right )+6 \left (\sqrt {x^4}-x^2\right ) \log \left (16 x^2 \sqrt {x^2 y(x)^{4/3}-4 y(x)^{5/3}}+x^6 \left (-\sqrt {x^2-4 \sqrt [3]{y(x)}}\right )-\sqrt {x^4} \sqrt {x^2-4 \sqrt [3]{y(x)}} \left (x^4-16 y(x)^{2/3}\right )\right )+6 \left (\sqrt {x^4}+x^2\right ) \log \left (16 x^2 \sqrt {x^2 y(x)^{4/3}-4 y(x)^{5/3}}+x^6 \sqrt {x^2-4 \sqrt [3]{y(x)}}-\sqrt {x^4} \sqrt {x^2-4 \sqrt [3]{y(x)}} \left (x^4-16 y(x)^{2/3}\right )\right )}{48 x^2}-\frac {\sqrt {\left (x^2-4 \sqrt [3]{y(x)}\right ) y(x)^{4/3}} \log \left (\sqrt {x^2-4 \sqrt [3]{y(x)}}-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}} \log \left (\sqrt {x^2-4 \sqrt [3]{y(x)}}-x\right )}{\sqrt {x^2-4 \sqrt [3]{y(x)}} y(x)^{2/3}}+\frac {8 x^2 \log (y(x))+6 \sqrt {x^4} \log \left (x^2 \sqrt {x^2-4 \sqrt [3]{y(x)}}\right )+3 \sqrt {x^4} \log \left (4 \sqrt [3]{y(x)}-x^2\right )+6 \left (x^2-\sqrt {x^4}\right ) \log \left (16 x^2 \sqrt {x^2 y(x)^{4/3}-4 y(x)^{5/3}}+x^6 \left (-\sqrt {x^2-4 \sqrt [3]{y(x)}}\right )-\sqrt {x^4} \sqrt {x^2-4 \sqrt [3]{y(x)}} \left (x^4-16 y(x)^{2/3}\right )\right )-6 \left (\sqrt {x^4}+x^2\right ) \log \left (16 x^2 \sqrt {x^2 y(x)^{4/3}-4 y(x)^{5/3}}+x^6 \sqrt {x^2-4 \sqrt [3]{y(x)}}-\sqrt {x^4} \sqrt {x^2-4 \sqrt [3]{y(x)}} \left (x^4-16 y(x)^{2/3}\right )\right )}{48 x^2}=c_1,y(x)\right ] \\ y(x)\to 0 \\ \end{align*}