60.1.535 problem 538

Internal problem ID [10549]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, linear first order
Problem number : 538
Date solved : Monday, January 27, 2025 at 08:54:26 PM
CAS classification : [[_homogeneous, `class G`]]

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

✓ Solution by Maple

Time used: 3.799 (sec). Leaf size: 1722

dsolve(2*(x*diff(y(x),x)+y(x))^3-y(x)*diff(y(x),x)=0,y(x), singsol=all)
 

\begin{align*} y &= 0 \\ 6^{{2}/{3}} \left (\int _{}^{y}\frac {\textit {\_f}^{2} \left (6^{{1}/{3}} x \textit {\_f} +{\left (-9 \left (-\frac {\sqrt {3}\, \sqrt {\frac {27 \textit {\_f}^{2} x -2 \textit {\_f}}{x}}}{9}+\textit {\_f} \right ) x^{2} \textit {\_f} \right )}^{{2}/{3}}\right ) \left (\int _{\textit {\_b}}^{x}\frac {\textit {\_a} \left (-18 \,3^{{5}/{6}} 2^{{1}/{3}} \textit {\_a}^{2} \textit {\_f}^{2}+3^{{5}/{6}} 2^{{1}/{3}} \textit {\_a} \textit {\_f} +6 \sqrt {\frac {27 \textit {\_f}^{2} \textit {\_a} -2 \textit {\_f}}{\textit {\_a}}}\, 2^{{1}/{3}} 3^{{1}/{3}} \textit {\_a}^{2} \textit {\_f} -9 {\left (-9 \left (-\frac {\sqrt {3}\, \sqrt {\frac {27 \textit {\_f}^{2} \textit {\_a} -2 \textit {\_f}}{\textit {\_a}}}}{9}+\textit {\_f} \right ) \textit {\_a}^{2} \textit {\_f} \right )}^{{2}/{3}} \sqrt {3}\, \textit {\_a} \textit {\_f} +3 \textit {\_a} {\left (-9 \left (-\frac {\sqrt {3}\, \sqrt {\frac {27 \textit {\_f}^{2} \textit {\_a} -2 \textit {\_f}}{\textit {\_a}}}}{9}+\textit {\_f} \right ) \textit {\_a}^{2} \textit {\_f} \right )}^{{2}/{3}} \sqrt {\frac {27 \textit {\_f}^{2} \textit {\_a} -2 \textit {\_f}}{\textit {\_a}}}+{\left (-9 \left (-\frac {\sqrt {3}\, \sqrt {\frac {27 \textit {\_f}^{2} \textit {\_a} -2 \textit {\_f}}{\textit {\_a}}}}{9}+\textit {\_f} \right ) \textit {\_a}^{2} \textit {\_f} \right )}^{{2}/{3}} \sqrt {3}\right )}{{\left (-9 \left (-\frac {\sqrt {3}\, \sqrt {\frac {27 \textit {\_f}^{2} \textit {\_a} -2 \textit {\_f}}{\textit {\_a}}}}{9}+\textit {\_f} \right ) \textit {\_a}^{2} \textit {\_f} \right )}^{{2}/{3}} \sqrt {\frac {27 \textit {\_f}^{2} \textit {\_a} -2 \textit {\_f}}{\textit {\_a}}}\, \left (2^{{1}/{3}} 3^{{1}/{3}} \textit {\_a} \textit {\_f} +{\left (-9 \left (-\frac {\sqrt {3}\, \sqrt {\frac {27 \textit {\_f}^{2} \textit {\_a} -2 \textit {\_f}}{\textit {\_a}}}}{9}+\textit {\_f} \right ) \textit {\_a}^{2} \textit {\_f} \right )}^{{2}/{3}}\right )^{2}}d \textit {\_a} \right )+x {\left (-9 \left (-\frac {\sqrt {3}\, \sqrt {\frac {27 \textit {\_f}^{2} x -2 \textit {\_f}}{x}}}{9}+\textit {\_f} \right ) x^{2} \textit {\_f} \right )}^{{1}/{3}}}{6^{{1}/{3}} x \textit {\_f} +{\left (-9 \left (-\frac {\sqrt {3}\, \sqrt {\frac {27 \textit {\_f}^{2} x -2 \textit {\_f}}{x}}}{9}+\textit {\_f} \right ) x^{2} \textit {\_f} \right )}^{{2}/{3}}}d \textit {\_f} \right )-\int _{\textit {\_b}}^{x}\frac {-6^{{2}/{3}} y {\left (-9 y \left (-\frac {\sqrt {3}\, \sqrt {\frac {27 y^{2} \textit {\_a} -2 y}{\textit {\_a}}}}{9}+y\right ) \textit {\_a}^{2}\right )}^{{1}/{3}} \textit {\_a} +6^{{1}/{3}} \textit {\_a} y+{\left (-9 y \left (-\frac {\sqrt {3}\, \sqrt {\frac {27 y^{2} \textit {\_a} -2 y}{\textit {\_a}}}}{9}+y\right ) \textit {\_a}^{2}\right )}^{{2}/{3}}}{\textit {\_a} \left (6^{{1}/{3}} \textit {\_a} y+{\left (-9 y \left (-\frac {\sqrt {3}\, \sqrt {\frac {27 y^{2} \textit {\_a} -2 y}{\textit {\_a}}}}{9}+y\right ) \textit {\_a}^{2}\right )}^{{2}/{3}}\right )}d \textit {\_a} +c_{1} &= 0 \\ 2 \,6^{{2}/{3}} \left (\int _{}^{y}\frac {\textit {\_f}^{2} \left (\left (-i \sqrt {3}-1\right ) {\left (-9 \left (-\frac {\sqrt {3}\, \sqrt {\frac {27 \textit {\_f}^{2} x -2 \textit {\_f}}{x}}}{9}+\textit {\_f} \right ) x^{2} \textit {\_f} \right )}^{{2}/{3}}+\textit {\_f} x 2^{{1}/{3}} \left (i 3^{{5}/{6}}-3^{{1}/{3}}\right )\right ) \left (\int _{\textit {\_b}}^{x}\frac {54 \left (\frac {\left (\frac {\textit {\_a} \left (1+i \sqrt {3}\right ) \sqrt {\frac {27 \textit {\_f}^{2} \textit {\_a} -2 \textit {\_f}}{\textit {\_a}}}}{9}+\left (-i-\frac {\sqrt {3}}{3}\right ) \left (\textit {\_a} \textit {\_f} -\frac {1}{9}\right )\right ) {\left (-9 \left (-\frac {\sqrt {3}\, \sqrt {\frac {27 \textit {\_f}^{2} \textit {\_a} -2 \textit {\_f}}{\textit {\_a}}}}{9}+\textit {\_f} \right ) \textit {\_a}^{2} \textit {\_f} \right )}^{{2}/{3}}}{2}+\left (\frac {\left (-i 3^{{5}/{6}}+3^{{1}/{3}}\right ) \textit {\_a} \sqrt {\frac {27 \textit {\_f}^{2} \textit {\_a} -2 \textit {\_f}}{\textit {\_a}}}}{9}+\left (\textit {\_a} \textit {\_f} -\frac {1}{18}\right ) \left (i 3^{{1}/{3}}-\frac {3^{{5}/{6}}}{3}\right )\right ) 2^{{1}/{3}} \textit {\_a} \textit {\_f} \right ) \textit {\_a}}{\sqrt {\frac {27 \textit {\_f}^{2} \textit {\_a} -2 \textit {\_f}}{\textit {\_a}}}\, {\left (-9 \left (-\frac {\sqrt {3}\, \sqrt {\frac {27 \textit {\_f}^{2} \textit {\_a} -2 \textit {\_f}}{\textit {\_a}}}}{9}+\textit {\_f} \right ) \textit {\_a}^{2} \textit {\_f} \right )}^{{2}/{3}} {\left (\left (i-\sqrt {3}\right ) {\left (-9 \left (-\frac {\sqrt {3}\, \sqrt {\frac {27 \textit {\_f}^{2} \textit {\_a} -2 \textit {\_f}}{\textit {\_a}}}}{9}+\textit {\_f} \right ) \textit {\_a}^{2} \textit {\_f} \right )}^{{2}/{3}}+2^{{1}/{3}} \textit {\_a} \left (3^{{5}/{6}}+i 3^{{1}/{3}}\right ) \textit {\_f} \right )}^{2}}d \textit {\_a} \right )+x {\left (-9 \left (-\frac {\sqrt {3}\, \sqrt {\frac {27 \textit {\_f}^{2} x -2 \textit {\_f}}{x}}}{9}+\textit {\_f} \right ) x^{2} \textit {\_f} \right )}^{{1}/{3}}}{\left (-i \sqrt {3}-1\right ) {\left (-9 \left (-\frac {\sqrt {3}\, \sqrt {\frac {27 \textit {\_f}^{2} x -2 \textit {\_f}}{x}}}{9}+\textit {\_f} \right ) x^{2} \textit {\_f} \right )}^{{2}/{3}}+\textit {\_f} x 2^{{1}/{3}} \left (i 3^{{5}/{6}}-3^{{1}/{3}}\right )}d \textit {\_f} \right )-\int _{\textit {\_b}}^{x}\frac {\left (-i+\sqrt {3}\right ) {\left (-9 y \left (-\frac {\sqrt {3}\, \sqrt {\frac {27 y^{2} \textit {\_a} -2 y}{\textit {\_a}}}}{9}+y\right ) \textit {\_a}^{2}\right )}^{{2}/{3}}-y \textit {\_a} \left (2 i {\left (-9 y \left (-\frac {\sqrt {3}\, \sqrt {\frac {27 y^{2} \textit {\_a} -2 y}{\textit {\_a}}}}{9}+y\right ) \textit {\_a}^{2}\right )}^{{1}/{3}} 2^{{2}/{3}} 3^{{2}/{3}}+2^{{1}/{3}} \left (3^{{5}/{6}}+i 3^{{1}/{3}}\right )\right )}{\left (\left (-i+\sqrt {3}\right ) {\left (-9 y \left (-\frac {\sqrt {3}\, \sqrt {\frac {27 y^{2} \textit {\_a} -2 y}{\textit {\_a}}}}{9}+y\right ) \textit {\_a}^{2}\right )}^{{2}/{3}}-y 2^{{1}/{3}} \textit {\_a} \left (3^{{5}/{6}}+i 3^{{1}/{3}}\right )\right ) \textit {\_a}}d \textit {\_a} +c_{1} &= 0 \\ -2 \,6^{{2}/{3}} \left (\int _{}^{y}\frac {\textit {\_f}^{2} \left (\int _{\textit {\_b}}^{x}\frac {54 \textit {\_a} \left (\left (\frac {\left (i \sqrt {3}-1\right ) \textit {\_a} \sqrt {\frac {27 \textit {\_f}^{2} \textit {\_a} -2 \textit {\_f}}{\textit {\_a}}}}{18}-\frac {\left (i-\frac {\sqrt {3}}{3}\right ) \left (\textit {\_a} \textit {\_f} -\frac {1}{9}\right )}{2}\right ) {\left (-9 \left (-\frac {\sqrt {3}\, \sqrt {\frac {27 \textit {\_f}^{2} \textit {\_a} -2 \textit {\_f}}{\textit {\_a}}}}{9}+\textit {\_f} \right ) \textit {\_a}^{2} \textit {\_f} \right )}^{{2}/{3}}+2^{{1}/{3}} \left (-\frac {\left (3^{{1}/{3}}+i 3^{{5}/{6}}\right ) \textit {\_a} \sqrt {\frac {27 \textit {\_f}^{2} \textit {\_a} -2 \textit {\_f}}{\textit {\_a}}}}{9}+\left (\textit {\_a} \textit {\_f} -\frac {1}{18}\right ) \left (i 3^{{1}/{3}}+\frac {3^{{5}/{6}}}{3}\right )\right ) \textit {\_a} \textit {\_f} \right )}{\sqrt {\frac {27 \textit {\_f}^{2} \textit {\_a} -2 \textit {\_f}}{\textit {\_a}}}\, {\left (-9 \left (-\frac {\sqrt {3}\, \sqrt {\frac {27 \textit {\_f}^{2} \textit {\_a} -2 \textit {\_f}}{\textit {\_a}}}}{9}+\textit {\_f} \right ) \textit {\_a}^{2} \textit {\_f} \right )}^{{2}/{3}} {\left (\left (\sqrt {3}+i\right ) {\left (-9 \left (-\frac {\sqrt {3}\, \sqrt {\frac {27 \textit {\_f}^{2} \textit {\_a} -2 \textit {\_f}}{\textit {\_a}}}}{9}+\textit {\_f} \right ) \textit {\_a}^{2} \textit {\_f} \right )}^{{2}/{3}}+2^{{1}/{3}} \textit {\_a} \textit {\_f} \left (-3^{{5}/{6}}+i 3^{{1}/{3}}\right )\right )}^{2}}d \textit {\_a} \right ) \left ({\left (-9 \left (-\frac {\sqrt {3}\, \sqrt {\frac {27 \textit {\_f}^{2} x -2 \textit {\_f}}{x}}}{9}+\textit {\_f} \right ) x^{2} \textit {\_f} \right )}^{{2}/{3}} \left (\sqrt {3}+i\right )+\left (-3^{{5}/{6}}+i 3^{{1}/{3}}\right ) \textit {\_f} x 2^{{1}/{3}}\right )+i x {\left (-9 \left (-\frac {\sqrt {3}\, \sqrt {\frac {27 \textit {\_f}^{2} x -2 \textit {\_f}}{x}}}{9}+\textit {\_f} \right ) x^{2} \textit {\_f} \right )}^{{1}/{3}}}{{\left (-9 \left (-\frac {\sqrt {3}\, \sqrt {\frac {27 \textit {\_f}^{2} x -2 \textit {\_f}}{x}}}{9}+\textit {\_f} \right ) x^{2} \textit {\_f} \right )}^{{2}/{3}} \left (\sqrt {3}+i\right )+\left (-3^{{5}/{6}}+i 3^{{1}/{3}}\right ) \textit {\_f} x 2^{{1}/{3}}}d \textit {\_f} \right )-\int _{\textit {\_b}}^{x}\frac {\left (i \sqrt {3}-1\right ) {\left (-9 y \left (-\frac {\sqrt {3}\, \sqrt {\frac {27 y^{2} \textit {\_a} -2 y}{\textit {\_a}}}}{9}+y\right ) \textit {\_a}^{2}\right )}^{{2}/{3}}-y \left (2 {\left (-9 y \left (-\frac {\sqrt {3}\, \sqrt {\frac {27 y^{2} \textit {\_a} -2 y}{\textit {\_a}}}}{9}+y\right ) \textit {\_a}^{2}\right )}^{{1}/{3}} 2^{{2}/{3}} 3^{{2}/{3}}+2^{{1}/{3}} \left (3^{{1}/{3}}+i 3^{{5}/{6}}\right )\right ) \textit {\_a}}{\textit {\_a} \left (\left (i \sqrt {3}-1\right ) {\left (-9 y \left (-\frac {\sqrt {3}\, \sqrt {\frac {27 y^{2} \textit {\_a} -2 y}{\textit {\_a}}}}{9}+y\right ) \textit {\_a}^{2}\right )}^{{2}/{3}}-y 2^{{1}/{3}} \textit {\_a} \left (3^{{1}/{3}}+i 3^{{5}/{6}}\right )\right )}d \textit {\_a} +c_{1} &= 0 \\ \end{align*}

✓ Solution by Mathematica

Time used: 62.971 (sec). Leaf size: 96

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

\[ y(x)\to \frac {\int _1^x\frac {\text {InverseFunction}\left [\log (\text {$\#$1})+\frac {3 \sqrt {\text {$\#$1}^2-8 \text {$\#$1}^3}}{\text {$\#$1}}+\frac {2 \arctan \left (\sqrt {8 \text {$\#$1}-1}\right ) \text {$\#$1} \sqrt {8 \text {$\#$1}-1}}{\sqrt {\text {$\#$1}^2-8 \text {$\#$1}^3}}\&\right ][c_1+2 \log (K[1])]}{K[1]}dK[1]}{x} \]