Internal problem ID [12162]
Book: Differential equations and the calculus of variations by L. ElSGOLTS. MIR PUBLISHERS,
MOSCOW, Third printing 1977.
Section: Chapter 1, First-Order Differential Equations. Problems page 88
Problem number: Problem 65.
ODE order: 1.
ODE degree: 2.
CAS Maple gives this as type [[_1st_order, _with_linear_symmetries], _dAlembert]
\[ \boxed {{y^{\prime }}^{2}-2 y^{\prime } x +y=0} \]
✓ Solution by Maple
Time used: 0.062 (sec). Leaf size: 611
dsolve(diff(y(x),x)^2-2*x*diff(y(x),x)+y(x)=0,y(x), singsol=all)
\begin{align*} y \left (x \right ) &= -\frac {\left (x^{2}+x \left (x^{3}+2 \sqrt {3}\, \sqrt {-c_{1} \left (x^{3}-3 c_{1} \right )}-6 c_{1} \right )^{\frac {1}{3}}+\left (x^{3}+2 \sqrt {3}\, \sqrt {-c_{1} \left (x^{3}-3 c_{1} \right )}-6 c_{1} \right )^{\frac {2}{3}}\right ) \left (x^{2}-3 x \left (x^{3}+2 \sqrt {3}\, \sqrt {-c_{1} \left (x^{3}-3 c_{1} \right )}-6 c_{1} \right )^{\frac {1}{3}}+\left (x^{3}+2 \sqrt {3}\, \sqrt {-c_{1} \left (x^{3}-3 c_{1} \right )}-6 c_{1} \right )^{\frac {2}{3}}\right )}{4 \left (x^{3}+2 \sqrt {3}\, \sqrt {-c_{1} \left (x^{3}-3 c_{1} \right )}-6 c_{1} \right )^{\frac {2}{3}}} \\ y \left (x \right ) &= -\frac {\left (i \sqrt {3}\, \left (x^{3}+2 \sqrt {3}\, \sqrt {-c_{1} \left (x^{3}-3 c_{1} \right )}-6 c_{1} \right )^{\frac {2}{3}}-i \sqrt {3}\, x^{2}+\left (x^{3}+2 \sqrt {3}\, \sqrt {-c_{1} \left (x^{3}-3 c_{1} \right )}-6 c_{1} \right )^{\frac {2}{3}}-2 x \left (x^{3}+2 \sqrt {3}\, \sqrt {-c_{1} \left (x^{3}-3 c_{1} \right )}-6 c_{1} \right )^{\frac {1}{3}}+x^{2}\right ) \left (i \sqrt {3}\, \left (x^{3}+2 \sqrt {3}\, \sqrt {-c_{1} \left (x^{3}-3 c_{1} \right )}-6 c_{1} \right )^{\frac {2}{3}}-i \sqrt {3}\, x^{2}+\left (x^{3}+2 \sqrt {3}\, \sqrt {-c_{1} \left (x^{3}-3 c_{1} \right )}-6 c_{1} \right )^{\frac {2}{3}}+6 x \left (x^{3}+2 \sqrt {3}\, \sqrt {-c_{1} \left (x^{3}-3 c_{1} \right )}-6 c_{1} \right )^{\frac {1}{3}}+x^{2}\right )}{16 \left (x^{3}+2 \sqrt {3}\, \sqrt {-c_{1} \left (x^{3}-3 c_{1} \right )}-6 c_{1} \right )^{\frac {2}{3}}} \\ y \left (x \right ) &= -\frac {\left (i \sqrt {3}\, x^{2}-i \sqrt {3}\, \left (x^{3}+2 \sqrt {3}\, \sqrt {-c_{1} \left (x^{3}-3 c_{1} \right )}-6 c_{1} \right )^{\frac {2}{3}}+x^{2}-2 x \left (x^{3}+2 \sqrt {3}\, \sqrt {-c_{1} \left (x^{3}-3 c_{1} \right )}-6 c_{1} \right )^{\frac {1}{3}}+\left (x^{3}+2 \sqrt {3}\, \sqrt {-c_{1} \left (x^{3}-3 c_{1} \right )}-6 c_{1} \right )^{\frac {2}{3}}\right ) \left (i \sqrt {3}\, x^{2}-i \sqrt {3}\, \left (x^{3}+2 \sqrt {3}\, \sqrt {-c_{1} \left (x^{3}-3 c_{1} \right )}-6 c_{1} \right )^{\frac {2}{3}}+x^{2}+6 x \left (x^{3}+2 \sqrt {3}\, \sqrt {-c_{1} \left (x^{3}-3 c_{1} \right )}-6 c_{1} \right )^{\frac {1}{3}}+\left (x^{3}+2 \sqrt {3}\, \sqrt {-c_{1} \left (x^{3}-3 c_{1} \right )}-6 c_{1} \right )^{\frac {2}{3}}\right )}{16 \left (x^{3}+2 \sqrt {3}\, \sqrt {-c_{1} \left (x^{3}-3 c_{1} \right )}-6 c_{1} \right )^{\frac {2}{3}}} \\ \end{align*}
✓ Solution by Mathematica
Time used: 60.164 (sec). Leaf size: 954
DSolve[y'[x]^2-2*x*y'[x]+y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {1}{4} \left (x^2+\frac {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}}}+\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}{72} \left (18 x^2-\frac {9 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}}}+9 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}{72} \left (18 x^2+\frac {9 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}}}-9 \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 ) \\ 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}{4 \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}{72} \left (18 x^2+\frac {9 \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}}}+9 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}{72} \left (18 x^2+\frac {9 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}}}-9 \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*}