Internal problem ID [7961]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, linear first order
Problem number: 381.
ODE order: 1.
ODE degree: 2.
CAS Maple gives this as type [[_1st_order, _with_linear_symmetries], _dAlembert]
Solve \begin {gather*} \boxed {\left (y^{\prime }\right )^{2}-2 x y^{\prime }+y=0} \end {gather*}
✓ Solution by Maple
Time used: 0.201 (sec). Leaf size: 656
dsolve(diff(y(x),x)^2-2*x*diff(y(x),x)+y(x) = 0,y(x), singsol=all)
\begin{align*} y \relax (x ) = -\left (\frac {\left (-6 c_{1}+x^{3}+2 \sqrt {-3 c_{1} x^{3}+9 c_{1}^{2}}\right )^{\frac {1}{3}}}{2}+\frac {x^{2}}{2 \left (-6 c_{1}+x^{3}+2 \sqrt {-3 c_{1} x^{3}+9 c_{1}^{2}}\right )^{\frac {1}{3}}}+\frac {x}{2}\right )^{2}+2 \left (\frac {\left (-6 c_{1}+x^{3}+2 \sqrt {-3 c_{1} x^{3}+9 c_{1}^{2}}\right )^{\frac {1}{3}}}{2}+\frac {x^{2}}{2 \left (-6 c_{1}+x^{3}+2 \sqrt {-3 c_{1} x^{3}+9 c_{1}^{2}}\right )^{\frac {1}{3}}}+\frac {x}{2}\right ) x \\ y \relax (x ) = -\left (-\frac {\left (-6 c_{1}+x^{3}+2 \sqrt {-3 c_{1} x^{3}+9 c_{1}^{2}}\right )^{\frac {1}{3}}}{4}-\frac {x^{2}}{4 \left (-6 c_{1}+x^{3}+2 \sqrt {-3 c_{1} x^{3}+9 c_{1}^{2}}\right )^{\frac {1}{3}}}+\frac {x}{2}-\frac {i \sqrt {3}\, \left (\frac {\left (-6 c_{1}+x^{3}+2 \sqrt {-3 c_{1} x^{3}+9 c_{1}^{2}}\right )^{\frac {1}{3}}}{2}-\frac {x^{2}}{2 \left (-6 c_{1}+x^{3}+2 \sqrt {-3 c_{1} x^{3}+9 c_{1}^{2}}\right )^{\frac {1}{3}}}\right )}{2}\right )^{2}+2 \left (-\frac {\left (-6 c_{1}+x^{3}+2 \sqrt {-3 c_{1} x^{3}+9 c_{1}^{2}}\right )^{\frac {1}{3}}}{4}-\frac {x^{2}}{4 \left (-6 c_{1}+x^{3}+2 \sqrt {-3 c_{1} x^{3}+9 c_{1}^{2}}\right )^{\frac {1}{3}}}+\frac {x}{2}-\frac {i \sqrt {3}\, \left (\frac {\left (-6 c_{1}+x^{3}+2 \sqrt {-3 c_{1} x^{3}+9 c_{1}^{2}}\right )^{\frac {1}{3}}}{2}-\frac {x^{2}}{2 \left (-6 c_{1}+x^{3}+2 \sqrt {-3 c_{1} x^{3}+9 c_{1}^{2}}\right )^{\frac {1}{3}}}\right )}{2}\right ) x \\ y \relax (x ) = -\left (-\frac {\left (-6 c_{1}+x^{3}+2 \sqrt {-3 c_{1} x^{3}+9 c_{1}^{2}}\right )^{\frac {1}{3}}}{4}-\frac {x^{2}}{4 \left (-6 c_{1}+x^{3}+2 \sqrt {-3 c_{1} x^{3}+9 c_{1}^{2}}\right )^{\frac {1}{3}}}+\frac {x}{2}+\frac {i \sqrt {3}\, \left (\frac {\left (-6 c_{1}+x^{3}+2 \sqrt {-3 c_{1} x^{3}+9 c_{1}^{2}}\right )^{\frac {1}{3}}}{2}-\frac {x^{2}}{2 \left (-6 c_{1}+x^{3}+2 \sqrt {-3 c_{1} x^{3}+9 c_{1}^{2}}\right )^{\frac {1}{3}}}\right )}{2}\right )^{2}+2 \left (-\frac {\left (-6 c_{1}+x^{3}+2 \sqrt {-3 c_{1} x^{3}+9 c_{1}^{2}}\right )^{\frac {1}{3}}}{4}-\frac {x^{2}}{4 \left (-6 c_{1}+x^{3}+2 \sqrt {-3 c_{1} x^{3}+9 c_{1}^{2}}\right )^{\frac {1}{3}}}+\frac {x}{2}+\frac {i \sqrt {3}\, \left (\frac {\left (-6 c_{1}+x^{3}+2 \sqrt {-3 c_{1} x^{3}+9 c_{1}^{2}}\right )^{\frac {1}{3}}}{2}-\frac {x^{2}}{2 \left (-6 c_{1}+x^{3}+2 \sqrt {-3 c_{1} x^{3}+9 c_{1}^{2}}\right )^{\frac {1}{3}}}\right )}{2}\right ) x \\ \end{align*}
✓ Solution by Mathematica
Time used: 2.497 (sec). Leaf size: 540
DSolve[y[x] - 2*x*y'[x] + y'[x]^2==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \text {Root}\left [16 \text {$\#$1}^6-24 \text {$\#$1}^5 x^2+9 \text {$\#$1}^4 x^4-8 \text {$\#$1}^3 e^{6 c_1}+42 \text {$\#$1}^2 e^{6 c_1} x^2-48 \text {$\#$1} e^{6 c_1} x^4+16 e^{6 c_1} x^6+e^{12 c_1}\&,1\right ] \\ y(x)\to \text {Root}\left [16 \text {$\#$1}^6-24 \text {$\#$1}^5 x^2+9 \text {$\#$1}^4 x^4-8 \text {$\#$1}^3 e^{6 c_1}+42 \text {$\#$1}^2 e^{6 c_1} x^2-48 \text {$\#$1} e^{6 c_1} x^4+16 e^{6 c_1} x^6+e^{12 c_1}\&,2\right ] \\ y(x)\to \text {Root}\left [16 \text {$\#$1}^6-24 \text {$\#$1}^5 x^2+9 \text {$\#$1}^4 x^4-8 \text {$\#$1}^3 e^{6 c_1}+42 \text {$\#$1}^2 e^{6 c_1} x^2-48 \text {$\#$1} e^{6 c_1} x^4+16 e^{6 c_1} x^6+e^{12 c_1}\&,3\right ] \\ y(x)\to \text {Root}\left [16 \text {$\#$1}^6-24 \text {$\#$1}^5 x^2+9 \text {$\#$1}^4 x^4-8 \text {$\#$1}^3 e^{6 c_1}+42 \text {$\#$1}^2 e^{6 c_1} x^2-48 \text {$\#$1} e^{6 c_1} x^4+16 e^{6 c_1} x^6+e^{12 c_1}\&,4\right ] \\ y(x)\to \text {Root}\left [16 \text {$\#$1}^6-24 \text {$\#$1}^5 x^2+9 \text {$\#$1}^4 x^4-8 \text {$\#$1}^3 e^{6 c_1}+42 \text {$\#$1}^2 e^{6 c_1} x^2-48 \text {$\#$1} e^{6 c_1} x^4+16 e^{6 c_1} x^6+e^{12 c_1}\&,5\right ] \\ y(x)\to \text {Root}\left [16 \text {$\#$1}^6-24 \text {$\#$1}^5 x^2+9 \text {$\#$1}^4 x^4-8 \text {$\#$1}^3 e^{6 c_1}+42 \text {$\#$1}^2 e^{6 c_1} x^2-48 \text {$\#$1} e^{6 c_1} x^4+16 e^{6 c_1} x^6+e^{12 c_1}\&,6\right ] \\ y(x)\to 0 \\ \end{align*}