Internal problem ID [3784]
Book: Ordinary differential equations and their solutions. By George Moseley Murphy.
1960
Section: Various 36
Problem number: 1073.
ODE order: 1.
ODE degree: 3.
CAS Maple gives this as type [[_1st_order, _with_linear_symmetries], _dAlembert]
Solve \begin {gather*} \boxed {2 y \left (y^{\prime }\right )^{3}-3 y^{\prime } x +2 y=0} \end {gather*}
✓ Solution by Maple
Time used: 0.36 (sec). Leaf size: 735
dsolve(2*y(x)*diff(y(x),x)^3-3*x*diff(y(x),x)+2*y(x) = 0,y(x), singsol=all)
\begin{align*} y \relax (x ) = \frac {2^{\frac {2}{3}} x}{2} \\ y \relax (x ) = \left (-\frac {2^{\frac {2}{3}}}{4}-\frac {i \sqrt {3}\, 2^{\frac {2}{3}}}{4}\right ) x \\ y \relax (x ) = \left (-\frac {2^{\frac {2}{3}}}{4}+\frac {i \sqrt {3}\, 2^{\frac {2}{3}}}{4}\right ) x \\ y \relax (x ) = 0 \\ y \relax (x ) = \RootOf \left (-\ln \relax (x )+\int _{}^{\textit {\_Z}}\frac {-2 \left (\left (\sqrt {2}\, \sqrt {\frac {1}{\textit {\_a} \left (2 \textit {\_a}^{3}-1\right )}}\, \textit {\_a}^{2}+1\right ) \left (2 \textit {\_a}^{3}-1\right )^{2}\right )^{\frac {1}{3}} \textit {\_a}^{3}-2 \textit {\_a}^{3}+\left (\left (\sqrt {2}\, \sqrt {\frac {1}{\textit {\_a} \left (2 \textit {\_a}^{3}-1\right )}}\, \textit {\_a}^{2}+1\right ) \left (2 \textit {\_a}^{3}-1\right )^{2}\right )^{\frac {2}{3}}+\left (\left (\sqrt {2}\, \sqrt {\frac {1}{\textit {\_a} \left (2 \textit {\_a}^{3}-1\right )}}\, \textit {\_a}^{2}+1\right ) \left (2 \textit {\_a}^{3}-1\right )^{2}\right )^{\frac {1}{3}}+1}{\textit {\_a} \left (2 \textit {\_a}^{3}-1\right ) \left (\left (\sqrt {2}\, \sqrt {\frac {1}{\textit {\_a} \left (2 \textit {\_a}^{3}-1\right )}}\, \textit {\_a}^{2}+1\right ) \left (2 \textit {\_a}^{3}-1\right )^{2}\right )^{\frac {1}{3}}}d \textit {\_a} +c_{1}\right ) x \\ y \relax (x ) = \RootOf \left (-2 \ln \relax (x )+\int _{}^{\textit {\_Z}}\frac {2 i \sqrt {3}\, \textit {\_a}^{3}+i \left (\left (\sqrt {2}\, \sqrt {\frac {1}{\textit {\_a} \left (2 \textit {\_a}^{3}-1\right )}}\, \textit {\_a}^{2}+1\right ) \left (2 \textit {\_a}^{3}-1\right )^{2}\right )^{\frac {2}{3}} \sqrt {3}-4 \left (\left (\sqrt {2}\, \sqrt {\frac {1}{\textit {\_a} \left (2 \textit {\_a}^{3}-1\right )}}\, \textit {\_a}^{2}+1\right ) \left (2 \textit {\_a}^{3}-1\right )^{2}\right )^{\frac {1}{3}} \textit {\_a}^{3}+2 \textit {\_a}^{3}-\left (\left (\sqrt {2}\, \sqrt {\frac {1}{\textit {\_a} \left (2 \textit {\_a}^{3}-1\right )}}\, \textit {\_a}^{2}+1\right ) \left (2 \textit {\_a}^{3}-1\right )^{2}\right )^{\frac {2}{3}}-i \sqrt {3}+2 \left (\left (\sqrt {2}\, \sqrt {\frac {1}{\textit {\_a} \left (2 \textit {\_a}^{3}-1\right )}}\, \textit {\_a}^{2}+1\right ) \left (2 \textit {\_a}^{3}-1\right )^{2}\right )^{\frac {1}{3}}-1}{\textit {\_a} \left (2 \textit {\_a}^{3}-1\right ) \left (\left (\sqrt {2}\, \sqrt {\frac {1}{\textit {\_a} \left (2 \textit {\_a}^{3}-1\right )}}\, \textit {\_a}^{2}+1\right ) \left (2 \textit {\_a}^{3}-1\right )^{2}\right )^{\frac {1}{3}}}d \textit {\_a} +2 c_{1}\right ) x \\ y \relax (x ) = \RootOf \left (-2 \ln \relax (x )-\left (\int _{}^{\textit {\_Z}}\frac {2 i \sqrt {3}\, \textit {\_a}^{3}+i \left (\left (\sqrt {2}\, \sqrt {\frac {1}{\textit {\_a} \left (2 \textit {\_a}^{3}-1\right )}}\, \textit {\_a}^{2}+1\right ) \left (2 \textit {\_a}^{3}-1\right )^{2}\right )^{\frac {2}{3}} \sqrt {3}+4 \left (\left (\sqrt {2}\, \sqrt {\frac {1}{\textit {\_a} \left (2 \textit {\_a}^{3}-1\right )}}\, \textit {\_a}^{2}+1\right ) \left (2 \textit {\_a}^{3}-1\right )^{2}\right )^{\frac {1}{3}} \textit {\_a}^{3}-2 \textit {\_a}^{3}+\left (\left (\sqrt {2}\, \sqrt {\frac {1}{\textit {\_a} \left (2 \textit {\_a}^{3}-1\right )}}\, \textit {\_a}^{2}+1\right ) \left (2 \textit {\_a}^{3}-1\right )^{2}\right )^{\frac {2}{3}}-i \sqrt {3}-2 \left (\left (\sqrt {2}\, \sqrt {\frac {1}{\textit {\_a} \left (2 \textit {\_a}^{3}-1\right )}}\, \textit {\_a}^{2}+1\right ) \left (2 \textit {\_a}^{3}-1\right )^{2}\right )^{\frac {1}{3}}+1}{\textit {\_a} \left (2 \textit {\_a}^{3}-1\right ) \left (\left (\sqrt {2}\, \sqrt {\frac {1}{\textit {\_a} \left (2 \textit {\_a}^{3}-1\right )}}\, \textit {\_a}^{2}+1\right ) \left (2 \textit {\_a}^{3}-1\right )^{2}\right )^{\frac {1}{3}}}d \textit {\_a} \right )+2 c_{1}\right ) x \\ \end{align*}
✓ Solution by Mathematica
Time used: 155.89 (sec). Leaf size: 10331
DSolve[2 y[x] (y'[x])^3 -3 x y'[x]+2 y[x]==0,y[x],x,IncludeSingularSolutions -> True]
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