Internal problem ID [8067]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, linear first order
Problem number: 487.
ODE order: 1.
ODE degree: 2.
CAS Maple gives this as type [[_1st_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {y^{2} \left (y^{\prime }\right )^{2}-6 x^{3} y^{\prime }+4 y x^{2}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.578 (sec). Leaf size: 118
dsolve(y(x)^2*diff(y(x),x)^2-6*x^3*diff(y(x),x)+4*x^2*y(x) = 0,y(x), singsol=all)
\begin{align*} y \relax (x ) = \frac {18^{\frac {1}{3}} x^{\frac {4}{3}}}{2} \\ y \relax (x ) = \left (-\frac {18^{\frac {1}{3}} x^{\frac {1}{3}}}{4}-\frac {i \sqrt {3}\, 18^{\frac {1}{3}} x^{\frac {1}{3}}}{4}\right ) x \\ y \relax (x ) = \left (-\frac {18^{\frac {1}{3}} x^{\frac {1}{3}}}{4}+\frac {i \sqrt {3}\, 18^{\frac {1}{3}} x^{\frac {1}{3}}}{4}\right ) x \\ y \relax (x ) = 0 \\ y \relax (x ) = \RootOf \left (-\ln \relax (x )+\int _{}^{\textit {\_Z}}-\frac {3 \left (4 \textit {\_a}^{3}+3 \sqrt {-4 \textit {\_a}^{3}+9}-9\right )}{4 \textit {\_a} \left (4 \textit {\_a}^{3}-9\right )}d \textit {\_a} +c_{1}\right ) x^{\frac {4}{3}} \\ \end{align*}
✓ Solution by Mathematica
Time used: 1.651 (sec). Leaf size: 298
DSolve[4*x^2*y[x] - 6*x^3*y'[x] + y[x]^2*y'[x]^2==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} \text {Solve}\left [\frac {\sqrt {9 x^6-4 x^2 y(x)^3} \log \left (\sqrt {9 x^4-4 y(x)^3}+3 x^2\right )}{2 x \sqrt {9 x^4-4 y(x)^3}}-\frac {3}{4} \left (\frac {\sqrt {9 x^6-4 x^2 y(x)^3}}{x \sqrt {9 x^4-4 y(x)^3}}-1\right ) \log (y(x))=c_1,y(x)\right ] \\ \text {Solve}\left [\frac {3}{4} \left (\frac {\sqrt {9 x^6-4 x^2 y(x)^3}}{x \sqrt {9 x^4-4 y(x)^3}}+1\right ) \log (y(x))-\frac {\sqrt {9 x^6-4 x^2 y(x)^3} \log \left (\sqrt {9 x^4-4 y(x)^3}+3 x^2\right )}{2 x \sqrt {9 x^4-4 y(x)^3}}=c_1,y(x)\right ] \\ y(x)\to \left (-\frac {3}{2}\right )^{2/3} x^{4/3} \\ y(x)\to \left (\frac {3}{2}\right )^{2/3} x^{4/3} \\ y(x)\to -\sqrt [3]{-1} \left (\frac {3}{2}\right )^{2/3} x^{4/3} \\ \end{align*}