29.36.8 problem 1072

Internal problem ID [5635]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Various 36
Problem number : 1072
Date solved : Monday, January 27, 2025 at 12:43:30 PM
CAS classification : [[_1st_order, _with_linear_symmetries], _dAlembert]

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

✓ Solution by Maple

Time used: 0.309 (sec). Leaf size: 601

dsolve(y(x)*diff(y(x),x)^3-3*x*diff(y(x),x)+3*y(x) = 0,y(x), singsol=all)
 

\begin{align*} y \left (x \right ) &= 0 \\ y \left (x \right ) &= \operatorname {RootOf}\left (-2 \ln \left (x \right )+\int _{}^{\textit {\_Z}}\frac {-2 {\left (4 \sqrt {\frac {9 \textit {\_a}^{3}-4}{\textit {\_a}}}\, \textit {\_a}^{5}+12 \textit {\_a}^{6}-24 \textit {\_a}^{3}+8\right )}^{{1}/{3}} \textit {\_a}^{3}-8 \textit {\_a}^{3}+{\left (4 \sqrt {\frac {9 \textit {\_a}^{3}-4}{\textit {\_a}}}\, \textit {\_a}^{5}+12 \textit {\_a}^{6}-24 \textit {\_a}^{3}+8\right )}^{{2}/{3}}+2 {\left (4 \sqrt {\frac {9 \textit {\_a}^{3}-4}{\textit {\_a}}}\, \textit {\_a}^{5}+12 \textit {\_a}^{6}-24 \textit {\_a}^{3}+8\right )}^{{1}/{3}}+4}{\textit {\_a}^{4} {\left (4 \sqrt {\frac {9 \textit {\_a}^{3}-4}{\textit {\_a}}}\, \textit {\_a}^{5}+12 \textit {\_a}^{6}-24 \textit {\_a}^{3}+8\right )}^{{1}/{3}}}d \textit {\_a} +2 c_{1} \right ) x \\ y \left (x \right ) &= \operatorname {RootOf}\left (-4 \ln \left (x \right )+\int _{}^{\textit {\_Z}}\frac {8 i \sqrt {3}\, \textit {\_a}^{3}+i \sqrt {3}\, {\left (4 \sqrt {\frac {9 \textit {\_a}^{3}-4}{\textit {\_a}}}\, \textit {\_a}^{5}+12 \textit {\_a}^{6}-24 \textit {\_a}^{3}+8\right )}^{{2}/{3}}-4 {\left (4 \sqrt {\frac {9 \textit {\_a}^{3}-4}{\textit {\_a}}}\, \textit {\_a}^{5}+12 \textit {\_a}^{6}-24 \textit {\_a}^{3}+8\right )}^{{1}/{3}} \textit {\_a}^{3}+8 \textit {\_a}^{3}-4 i \sqrt {3}-{\left (4 \sqrt {\frac {9 \textit {\_a}^{3}-4}{\textit {\_a}}}\, \textit {\_a}^{5}+12 \textit {\_a}^{6}-24 \textit {\_a}^{3}+8\right )}^{{2}/{3}}+4 {\left (4 \sqrt {\frac {9 \textit {\_a}^{3}-4}{\textit {\_a}}}\, \textit {\_a}^{5}+12 \textit {\_a}^{6}-24 \textit {\_a}^{3}+8\right )}^{{1}/{3}}-4}{\textit {\_a}^{4} {\left (4 \sqrt {\frac {9 \textit {\_a}^{3}-4}{\textit {\_a}}}\, \textit {\_a}^{5}+12 \textit {\_a}^{6}-24 \textit {\_a}^{3}+8\right )}^{{1}/{3}}}d \textit {\_a} +4 c_{1} \right ) x \\ y \left (x \right ) &= \operatorname {RootOf}\left (-4 \ln \left (x \right )-\int _{}^{\textit {\_Z}}\frac {8 i \sqrt {3}\, \textit {\_a}^{3}+i \sqrt {3}\, {\left (4 \sqrt {\frac {9 \textit {\_a}^{3}-4}{\textit {\_a}}}\, \textit {\_a}^{5}+12 \textit {\_a}^{6}-24 \textit {\_a}^{3}+8\right )}^{{2}/{3}}+4 {\left (4 \sqrt {\frac {9 \textit {\_a}^{3}-4}{\textit {\_a}}}\, \textit {\_a}^{5}+12 \textit {\_a}^{6}-24 \textit {\_a}^{3}+8\right )}^{{1}/{3}} \textit {\_a}^{3}-8 \textit {\_a}^{3}-4 i \sqrt {3}+{\left (4 \sqrt {\frac {9 \textit {\_a}^{3}-4}{\textit {\_a}}}\, \textit {\_a}^{5}+12 \textit {\_a}^{6}-24 \textit {\_a}^{3}+8\right )}^{{2}/{3}}-4 {\left (4 \sqrt {\frac {9 \textit {\_a}^{3}-4}{\textit {\_a}}}\, \textit {\_a}^{5}+12 \textit {\_a}^{6}-24 \textit {\_a}^{3}+8\right )}^{{1}/{3}}+4}{\textit {\_a}^{4} {\left (4 \sqrt {\frac {9 \textit {\_a}^{3}-4}{\textit {\_a}}}\, \textit {\_a}^{5}+12 \textit {\_a}^{6}-24 \textit {\_a}^{3}+8\right )}^{{1}/{3}}}d \textit {\_a} +4 c_{1} \right ) x \\ \end{align*}

✓ Solution by Mathematica

Time used: 112.235 (sec). Leaf size: 8706

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

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