Internal problem ID [8717]
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]
\[ \boxed {{y^{\prime }}^{2}-2 x y^{\prime }+y=0} \]
✓ Solution by Maple
Time used: 0.063 (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 \left (x^{3}+2 \sqrt {3}\, \sqrt {-c_{1} \left (x^{3}-3 c_{1} \right )}-6 c_{1} \right )^{\frac {2}{3}} \sqrt {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 \left (x^{3}+2 \sqrt {3}\, \sqrt {-c_{1} \left (x^{3}-3 c_{1} \right )}-6 c_{1} \right )^{\frac {2}{3}} \sqrt {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 \left (x^{3}+2 \sqrt {3}\, \sqrt {-c_{1} \left (x^{3}-3 c_{1} \right )}-6 c_{1} \right )^{\frac {2}{3}} \sqrt {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 \left (x^{3}+2 \sqrt {3}\, \sqrt {-c_{1} \left (x^{3}-3 c_{1} \right )}-6 c_{1} \right )^{\frac {2}{3}} \sqrt {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: 60.176 (sec). Leaf size: 954
DSolve[y[x] - 2*x*y'[x] + y'[x]^2==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*}