Internal problem ID [3566]
Book: Ordinary differential equations and their solutions. By George Moseley Murphy.
1960
Section: Various 29
Problem number: 835.
ODE order: 1.
ODE degree: 2.
CAS Maple gives this as type [[_1st_order, _with_linear_symmetries], _dAlembert]
Solve \begin {gather*} \boxed {3 \left (y^{\prime }\right )^{2}-2 y^{\prime } x +y=0} \end {gather*}
✓ Solution by Maple
Time used: 0.187 (sec). Leaf size: 656
dsolve(3*diff(y(x),x)^2-2*x*diff(y(x),x)+y(x) = 0,y(x), singsol=all)
\begin{align*} y \relax (x ) = -3 \left (\frac {\left (-54 c_{1}+x^{3}+6 \sqrt {-3 x^{3} c_{1}+81 c_{1}^{2}}\right )^{\frac {1}{3}}}{6}+\frac {x^{2}}{6 \left (-54 c_{1}+x^{3}+6 \sqrt {-3 x^{3} c_{1}+81 c_{1}^{2}}\right )^{\frac {1}{3}}}+\frac {x}{6}\right )^{2}+2 \left (\frac {\left (-54 c_{1}+x^{3}+6 \sqrt {-3 x^{3} c_{1}+81 c_{1}^{2}}\right )^{\frac {1}{3}}}{6}+\frac {x^{2}}{6 \left (-54 c_{1}+x^{3}+6 \sqrt {-3 x^{3} c_{1}+81 c_{1}^{2}}\right )^{\frac {1}{3}}}+\frac {x}{6}\right ) x \\ y \relax (x ) = -3 \left (-\frac {\left (-54 c_{1}+x^{3}+6 \sqrt {-3 x^{3} c_{1}+81 c_{1}^{2}}\right )^{\frac {1}{3}}}{12}-\frac {x^{2}}{12 \left (-54 c_{1}+x^{3}+6 \sqrt {-3 x^{3} c_{1}+81 c_{1}^{2}}\right )^{\frac {1}{3}}}+\frac {x}{6}-\frac {i \sqrt {3}\, \left (\frac {\left (-54 c_{1}+x^{3}+6 \sqrt {-3 x^{3} c_{1}+81 c_{1}^{2}}\right )^{\frac {1}{3}}}{6}-\frac {x^{2}}{6 \left (-54 c_{1}+x^{3}+6 \sqrt {-3 x^{3} c_{1}+81 c_{1}^{2}}\right )^{\frac {1}{3}}}\right )}{2}\right )^{2}+2 \left (-\frac {\left (-54 c_{1}+x^{3}+6 \sqrt {-3 x^{3} c_{1}+81 c_{1}^{2}}\right )^{\frac {1}{3}}}{12}-\frac {x^{2}}{12 \left (-54 c_{1}+x^{3}+6 \sqrt {-3 x^{3} c_{1}+81 c_{1}^{2}}\right )^{\frac {1}{3}}}+\frac {x}{6}-\frac {i \sqrt {3}\, \left (\frac {\left (-54 c_{1}+x^{3}+6 \sqrt {-3 x^{3} c_{1}+81 c_{1}^{2}}\right )^{\frac {1}{3}}}{6}-\frac {x^{2}}{6 \left (-54 c_{1}+x^{3}+6 \sqrt {-3 x^{3} c_{1}+81 c_{1}^{2}}\right )^{\frac {1}{3}}}\right )}{2}\right ) x \\ y \relax (x ) = -3 \left (-\frac {\left (-54 c_{1}+x^{3}+6 \sqrt {-3 x^{3} c_{1}+81 c_{1}^{2}}\right )^{\frac {1}{3}}}{12}-\frac {x^{2}}{12 \left (-54 c_{1}+x^{3}+6 \sqrt {-3 x^{3} c_{1}+81 c_{1}^{2}}\right )^{\frac {1}{3}}}+\frac {x}{6}+\frac {i \sqrt {3}\, \left (\frac {\left (-54 c_{1}+x^{3}+6 \sqrt {-3 x^{3} c_{1}+81 c_{1}^{2}}\right )^{\frac {1}{3}}}{6}-\frac {x^{2}}{6 \left (-54 c_{1}+x^{3}+6 \sqrt {-3 x^{3} c_{1}+81 c_{1}^{2}}\right )^{\frac {1}{3}}}\right )}{2}\right )^{2}+2 \left (-\frac {\left (-54 c_{1}+x^{3}+6 \sqrt {-3 x^{3} c_{1}+81 c_{1}^{2}}\right )^{\frac {1}{3}}}{12}-\frac {x^{2}}{12 \left (-54 c_{1}+x^{3}+6 \sqrt {-3 x^{3} c_{1}+81 c_{1}^{2}}\right )^{\frac {1}{3}}}+\frac {x}{6}+\frac {i \sqrt {3}\, \left (\frac {\left (-54 c_{1}+x^{3}+6 \sqrt {-3 x^{3} c_{1}+81 c_{1}^{2}}\right )^{\frac {1}{3}}}{6}-\frac {x^{2}}{6 \left (-54 c_{1}+x^{3}+6 \sqrt {-3 x^{3} c_{1}+81 c_{1}^{2}}\right )^{\frac {1}{3}}}\right )}{2}\right ) x \\ \end{align*}
✓ Solution by Mathematica
Time used: 60.166 (sec). Leaf size: 994
DSolve[3 (y'[x])^2-2 x y'[x]+y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {1}{12} \left (x^2+\frac {x \left (x^3+216 e^{3 c_1}\right )}{\sqrt [3]{x^6-540 e^{3 c_1} x^3+24 \sqrt {3} \sqrt {e^{3 c_1} \left (-x^3+27 e^{3 c_1}\right ){}^3}-5832 e^{6 c_1}}}+\sqrt [3]{x^6-540 e^{3 c_1} x^3+24 \sqrt {3} \sqrt {e^{3 c_1} \left (-x^3+27 e^{3 c_1}\right ){}^3}-5832 e^{6 c_1}}\right ) \\ y(x)\to \frac {1}{24} \left (2 x^2+\frac {\left (-1-i \sqrt {3}\right ) x \left (x^3+216 e^{3 c_1}\right )}{\sqrt [3]{x^6-540 e^{3 c_1} x^3+24 \sqrt {3} \sqrt {e^{3 c_1} \left (-x^3+27 e^{3 c_1}\right ){}^3}-5832 e^{6 c_1}}}+i \left (\sqrt {3}+i\right ) \sqrt [3]{x^6-540 e^{3 c_1} x^3+24 \sqrt {3} \sqrt {e^{3 c_1} \left (-x^3+27 e^{3 c_1}\right ){}^3}-5832 e^{6 c_1}}\right ) \\ y(x)\to \frac {1}{24} \left (2 x^2+\frac {i \left (\sqrt {3}+i\right ) x \left (x^3+216 e^{3 c_1}\right )}{\sqrt [3]{x^6-540 e^{3 c_1} x^3+24 \sqrt {3} \sqrt {e^{3 c_1} \left (-x^3+27 e^{3 c_1}\right ){}^3}-5832 e^{6 c_1}}}-\left (1+i \sqrt {3}\right ) \sqrt [3]{x^6-540 e^{3 c_1} x^3+24 \sqrt {3} \sqrt {e^{3 c_1} \left (-x^3+27 e^{3 c_1}\right ){}^3}-5832 e^{6 c_1}}\right ) \\ y(x)\to \frac {x^4+\left (x^6+20 e^{3 c_1} x^3+8 \sqrt {e^{3 c_1} \left (x^3+e^{3 c_1}\right ){}^3}-8 e^{6 c_1}\right ){}^{2/3}+x^2 \sqrt [3]{x^6+20 e^{3 c_1} x^3+8 \sqrt {e^{3 c_1} \left (x^3+e^{3 c_1}\right ){}^3}-8 e^{6 c_1}}-8 e^{3 c_1} x}{12 \sqrt [3]{x^6+20 e^{3 c_1} x^3+8 \sqrt {e^{3 c_1} \left (x^3+e^{3 c_1}\right ){}^3}-8 e^{6 c_1}}} \\ y(x)\to \frac {1}{24} \left (2 x^2+\frac {\left (1+i \sqrt {3}\right ) x \left (-x^3+8 e^{3 c_1}\right )}{\sqrt [3]{x^6+20 e^{3 c_1} x^3+8 \sqrt {e^{3 c_1} \left (x^3+e^{3 c_1}\right ){}^3}-8 e^{6 c_1}}}+i \left (\sqrt {3}+i\right ) \sqrt [3]{x^6+20 e^{3 c_1} x^3+8 \sqrt {e^{3 c_1} \left (x^3+e^{3 c_1}\right ){}^3}-8 e^{6 c_1}}\right ) \\ y(x)\to \frac {1}{24} \left (2 x^2+\frac {i \left (\sqrt {3}+i\right ) x \left (x^3-8 e^{3 c_1}\right )}{\sqrt [3]{x^6+20 e^{3 c_1} x^3+8 \sqrt {e^{3 c_1} \left (x^3+e^{3 c_1}\right ){}^3}-8 e^{6 c_1}}}-\left (1+i \sqrt {3}\right ) \sqrt [3]{x^6+20 e^{3 c_1} x^3+8 \sqrt {e^{3 c_1} \left (x^3+e^{3 c_1}\right ){}^3}-8 e^{6 c_1}}\right ) \\ \end{align*}