29.36.9 problem 1073

Internal problem ID [5636]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Various 36
Problem number : 1073
Date solved : Monday, January 27, 2025 at 12:43:38 PM
CAS classification : [[_1st_order, _with_linear_symmetries], _dAlembert]

\begin{align*} 2 y {y^{\prime }}^{3}-3 x y^{\prime }+2 y&=0 \end{align*}

✓ Solution by Maple

Time used: 0.135 (sec). Leaf size: 726

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 \left (x \right ) &= \frac {2^{{2}/{3}} x}{2} \\ y \left (x \right ) &= -\frac {2^{{2}/{3}} \left (1+i \sqrt {3}\right ) x}{4} \\ y \left (x \right ) &= \frac {2^{{2}/{3}} \left (i \sqrt {3}-1\right ) x}{4} \\ y \left (x \right ) &= 0 \\ y \left (x \right ) &= \operatorname {RootOf}\left (-\ln \left (x \right )+\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 )}^{{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 )}^{{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 )}^{{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 )}^{{1}/{3}}}d \textit {\_a} +c_{1} \right ) x \\ y \left (x \right ) &= \operatorname {RootOf}\left (-2 \ln \left (x \right )+\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 )}^{{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 )}^{{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 )}^{{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 )}^{{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 )}^{{1}/{3}}}d \textit {\_a} +2 c_{1} \right ) x \\ y \left (x \right ) &= \operatorname {RootOf}\left (-2 \ln \left (x \right )-\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 )}^{{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 )}^{{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 )}^{{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 )}^{{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 )}^{{1}/{3}}}d \textit {\_a} +2 c_{1} \right ) x \\ \end{align*}

✓ Solution by Mathematica

Time used: 166.564 (sec). Leaf size: 10331

DSolve[2 y[x] (D[y[x],x])^3 -3 x D[y[x],x]+2 y[x]==0,y[x],x,IncludeSingularSolutions -> True]
 

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