Internal problem ID [8000]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, linear first order
Problem number: 420.
ODE order: 1.
ODE degree: 2.
CAS Maple gives this as type [[_homogeneous, class G], _rational, _dAlembert]
Solve \begin {gather*} \boxed {x \left (y^{\prime }\right )^{2}-2 y y^{\prime }+a=0} \end {gather*}
✓ Solution by Maple
Time used: 0.223 (sec). Leaf size: 897
dsolve(x*diff(y(x),x)^2-2*y(x)*diff(y(x),x)+a = 0,y(x), singsol=all)
\begin{align*} y \relax (x ) = \frac {\left (\frac {\left (-36 a c_{1}^{2}+8 x^{3}+12 \sqrt {a \left (9 a c_{1}^{2}-4 x^{3}\right )}\, c_{1}\right )^{\frac {1}{3}}}{6 c_{1}}+\frac {2 x^{2}}{3 c_{1} \left (-36 a c_{1}^{2}+8 x^{3}+12 \sqrt {a \left (9 a c_{1}^{2}-4 x^{3}\right )}\, c_{1}\right )^{\frac {1}{3}}}+\frac {x}{3 c_{1}}\right ) x}{2}+\frac {a}{\frac {\left (-36 a c_{1}^{2}+8 x^{3}+12 \sqrt {a \left (9 a c_{1}^{2}-4 x^{3}\right )}\, c_{1}\right )^{\frac {1}{3}}}{3 c_{1}}+\frac {4 x^{2}}{3 c_{1} \left (-36 a c_{1}^{2}+8 x^{3}+12 \sqrt {a \left (9 a c_{1}^{2}-4 x^{3}\right )}\, c_{1}\right )^{\frac {1}{3}}}+\frac {2 x}{3 c_{1}}} \\ y \relax (x ) = \frac {\left (-\frac {\left (-36 a c_{1}^{2}+8 x^{3}+12 \sqrt {a \left (9 a c_{1}^{2}-4 x^{3}\right )}\, c_{1}\right )^{\frac {1}{3}}}{12 c_{1}}-\frac {x^{2}}{3 c_{1} \left (-36 a c_{1}^{2}+8 x^{3}+12 \sqrt {a \left (9 a c_{1}^{2}-4 x^{3}\right )}\, c_{1}\right )^{\frac {1}{3}}}+\frac {x}{3 c_{1}}-\frac {i \sqrt {3}\, \left (\frac {\left (-36 a c_{1}^{2}+8 x^{3}+12 \sqrt {a \left (9 a c_{1}^{2}-4 x^{3}\right )}\, c_{1}\right )^{\frac {1}{3}}}{6 c_{1}}-\frac {2 x^{2}}{3 c_{1} \left (-36 a c_{1}^{2}+8 x^{3}+12 \sqrt {a \left (9 a c_{1}^{2}-4 x^{3}\right )}\, c_{1}\right )^{\frac {1}{3}}}\right )}{2}\right ) x}{2}+\frac {a}{-\frac {\left (-36 a c_{1}^{2}+8 x^{3}+12 \sqrt {a \left (9 a c_{1}^{2}-4 x^{3}\right )}\, c_{1}\right )^{\frac {1}{3}}}{6 c_{1}}-\frac {2 x^{2}}{3 c_{1} \left (-36 a c_{1}^{2}+8 x^{3}+12 \sqrt {a \left (9 a c_{1}^{2}-4 x^{3}\right )}\, c_{1}\right )^{\frac {1}{3}}}+\frac {2 x}{3 c_{1}}-i \sqrt {3}\, \left (\frac {\left (-36 a c_{1}^{2}+8 x^{3}+12 \sqrt {a \left (9 a c_{1}^{2}-4 x^{3}\right )}\, c_{1}\right )^{\frac {1}{3}}}{6 c_{1}}-\frac {2 x^{2}}{3 c_{1} \left (-36 a c_{1}^{2}+8 x^{3}+12 \sqrt {a \left (9 a c_{1}^{2}-4 x^{3}\right )}\, c_{1}\right )^{\frac {1}{3}}}\right )} \\ y \relax (x ) = \frac {\left (-\frac {\left (-36 a c_{1}^{2}+8 x^{3}+12 \sqrt {a \left (9 a c_{1}^{2}-4 x^{3}\right )}\, c_{1}\right )^{\frac {1}{3}}}{12 c_{1}}-\frac {x^{2}}{3 c_{1} \left (-36 a c_{1}^{2}+8 x^{3}+12 \sqrt {a \left (9 a c_{1}^{2}-4 x^{3}\right )}\, c_{1}\right )^{\frac {1}{3}}}+\frac {x}{3 c_{1}}+\frac {i \sqrt {3}\, \left (\frac {\left (-36 a c_{1}^{2}+8 x^{3}+12 \sqrt {a \left (9 a c_{1}^{2}-4 x^{3}\right )}\, c_{1}\right )^{\frac {1}{3}}}{6 c_{1}}-\frac {2 x^{2}}{3 c_{1} \left (-36 a c_{1}^{2}+8 x^{3}+12 \sqrt {a \left (9 a c_{1}^{2}-4 x^{3}\right )}\, c_{1}\right )^{\frac {1}{3}}}\right )}{2}\right ) x}{2}+\frac {a}{-\frac {\left (-36 a c_{1}^{2}+8 x^{3}+12 \sqrt {a \left (9 a c_{1}^{2}-4 x^{3}\right )}\, c_{1}\right )^{\frac {1}{3}}}{6 c_{1}}-\frac {2 x^{2}}{3 c_{1} \left (-36 a c_{1}^{2}+8 x^{3}+12 \sqrt {a \left (9 a c_{1}^{2}-4 x^{3}\right )}\, c_{1}\right )^{\frac {1}{3}}}+\frac {2 x}{3 c_{1}}+i \sqrt {3}\, \left (\frac {\left (-36 a c_{1}^{2}+8 x^{3}+12 \sqrt {a \left (9 a c_{1}^{2}-4 x^{3}\right )}\, c_{1}\right )^{\frac {1}{3}}}{6 c_{1}}-\frac {2 x^{2}}{3 c_{1} \left (-36 a c_{1}^{2}+8 x^{3}+12 \sqrt {a \left (9 a c_{1}^{2}-4 x^{3}\right )}\, c_{1}\right )^{\frac {1}{3}}}\right )} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.101 (sec). Leaf size: 1550
DSolve[a - 2*y[x]*y'[x] + x*y'[x]^2==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {e^{-\frac {3 c_1}{2}} \left (a^4 x^4+\left (-a^6 x^6+20 a^3 e^{3 c_1} x^3+8 \sqrt {e^{3 c_1} \left (-a^3 x^3+e^{3 c_1}\right ){}^3}+8 e^{6 c_1}\right ){}^{2/3}-a^2 x^2 \sqrt [3]{-a^6 x^6+20 a^3 e^{3 c_1} x^3+8 \sqrt {e^{3 c_1} \left (-a^3 x^3+e^{3 c_1}\right ){}^3}+8 e^{6 c_1}}+8 a e^{3 c_1} x\right )}{4 \sqrt [3]{-a^6 x^6+20 a^3 e^{3 c_1} x^3+8 \sqrt {e^{3 c_1} \left (-a^3 x^3+e^{3 c_1}\right ){}^3}+8 e^{6 c_1}}} \\ y(x)\to \frac {i e^{-\frac {3 c_1}{2}} \left (-\left (\left (\sqrt {3}-i\right ) a^4 x^4\right )+\left (\sqrt {3}+i\right ) \left (-a^6 x^6+20 a^3 e^{3 c_1} x^3+8 \sqrt {e^{3 c_1} \left (-a^3 x^3+e^{3 c_1}\right ){}^3}+8 e^{6 c_1}\right ){}^{2/3}+2 i a^2 x^2 \sqrt [3]{-a^6 x^6+20 a^3 e^{3 c_1} x^3+8 \sqrt {e^{3 c_1} \left (-a^3 x^3+e^{3 c_1}\right ){}^3}+8 e^{6 c_1}}-8 \left (\sqrt {3}-i\right ) a e^{3 c_1} x\right )}{8 \sqrt [3]{-a^6 x^6+20 a^3 e^{3 c_1} x^3+8 \sqrt {e^{3 c_1} \left (-a^3 x^3+e^{3 c_1}\right ){}^3}+8 e^{6 c_1}}} \\ y(x)\to \frac {e^{-\frac {3 c_1}{2}} \left (i \left (\sqrt {3}+i\right ) a^4 x^4+\left (-1-i \sqrt {3}\right ) \left (-a^6 x^6+20 a^3 e^{3 c_1} x^3+8 \sqrt {e^{3 c_1} \left (-a^3 x^3+e^{3 c_1}\right ){}^3}+8 e^{6 c_1}\right ){}^{2/3}-2 a^2 x^2 \sqrt [3]{-a^6 x^6+20 a^3 e^{3 c_1} x^3+8 \sqrt {e^{3 c_1} \left (-a^3 x^3+e^{3 c_1}\right ){}^3}+8 e^{6 c_1}}+8 i \left (\sqrt {3}+i\right ) a e^{3 c_1} x\right )}{8 \sqrt [3]{-a^6 x^6+20 a^3 e^{3 c_1} x^3+8 \sqrt {e^{3 c_1} \left (-a^3 x^3+e^{3 c_1}\right ){}^3}+8 e^{6 c_1}}} \\ y(x)\to \frac {e^{-\frac {3 c_1}{2}} \left (a^4 x^4+\left (a^6 x^6-20 a^3 e^{3 c_1} x^3+8 \sqrt {e^{3 c_1} \left (-a^3 x^3+e^{3 c_1}\right ){}^3}-8 e^{6 c_1}\right ){}^{2/3}+a^2 x^2 \sqrt [3]{a^6 x^6-20 a^3 e^{3 c_1} x^3+8 \sqrt {e^{3 c_1} \left (-a^3 x^3+e^{3 c_1}\right ){}^3}-8 e^{6 c_1}}+8 a e^{3 c_1} x\right )}{4 \sqrt [3]{a^6 x^6-20 a^3 e^{3 c_1} x^3+8 \sqrt {e^{3 c_1} \left (-a^3 x^3+e^{3 c_1}\right ){}^3}-8 e^{6 c_1}}} \\ y(x)\to \frac {e^{-\frac {3 c_1}{2}} \left (\left (-1-i \sqrt {3}\right ) a^4 x^4+i \left (\sqrt {3}+i\right ) \left (a^6 x^6-20 a^3 e^{3 c_1} x^3+8 \sqrt {e^{3 c_1} \left (-a^3 x^3+e^{3 c_1}\right ){}^3}-8 e^{6 c_1}\right ){}^{2/3}+2 a^2 x^2 \sqrt [3]{a^6 x^6-20 a^3 e^{3 c_1} x^3+8 \sqrt {e^{3 c_1} \left (-a^3 x^3+e^{3 c_1}\right ){}^3}-8 e^{6 c_1}}+\left (-8-8 i \sqrt {3}\right ) a e^{3 c_1} x\right )}{8 \sqrt [3]{a^6 x^6-20 a^3 e^{3 c_1} x^3+8 \sqrt {e^{3 c_1} \left (-a^3 x^3+e^{3 c_1}\right ){}^3}-8 e^{6 c_1}}} \\ y(x)\to \frac {e^{-\frac {3 c_1}{2}} \left (i \left (\sqrt {3}+i\right ) a^4 x^4+\left (-1-i \sqrt {3}\right ) \left (a^6 x^6-20 a^3 e^{3 c_1} x^3+8 \sqrt {e^{3 c_1} \left (-a^3 x^3+e^{3 c_1}\right ){}^3}-8 e^{6 c_1}\right ){}^{2/3}+2 a^2 x^2 \sqrt [3]{a^6 x^6-20 a^3 e^{3 c_1} x^3+8 \sqrt {e^{3 c_1} \left (-a^3 x^3+e^{3 c_1}\right ){}^3}-8 e^{6 c_1}}+8 i \left (\sqrt {3}+i\right ) a e^{3 c_1} x\right )}{8 \sqrt [3]{a^6 x^6-20 a^3 e^{3 c_1} x^3+8 \sqrt {e^{3 c_1} \left (-a^3 x^3+e^{3 c_1}\right ){}^3}-8 e^{6 c_1}}} \\ \end{align*}