Internal problem ID [8801]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, linear first order
Problem number: 466.
ODE order: 1.
ODE degree: 2.
CAS Maple gives this as type [[_homogeneous, `class A`], _rational, _dAlembert]
\[ \boxed {y {y^{\prime }}^{2}-2 x y^{\prime }+y=0} \]
✓ Solution by Maple
Time used: 0.172 (sec). Leaf size: 71
dsolve(y(x)*diff(y(x),x)^2-2*x*diff(y(x),x)+y(x) = 0,y(x), singsol=all)
\begin{align*} y \left (x \right ) &= -x \\ y \left (x \right ) &= x \\ y \left (x \right ) &= 0 \\ y \left (x \right ) &= \sqrt {c_{1} \left (-2 i x +c_{1} \right )} \\ y \left (x \right ) &= \sqrt {c_{1} \left (2 i x +c_{1} \right )} \\ y \left (x \right ) &= -\sqrt {c_{1} \left (-2 i x +c_{1} \right )} \\ y \left (x \right ) &= -\sqrt {c_{1} \left (2 i x +c_{1} \right )} \\ \end{align*}
✓ Solution by Mathematica
Time used: 2.634 (sec). Leaf size: 174
DSolve[y[x] - 2*x*y'[x] + y[x]*y'[x]^2==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -\frac {1}{4} \left (\cosh \left (\frac {c_1}{2}\right )+\sinh \left (\frac {c_1}{2}\right )\right ) \sqrt {-8 i x+\cosh (c_1)+\sinh (c_1)} \\ y(x)\to \frac {1}{4} \left (\cosh \left (\frac {c_1}{2}\right )+\sinh \left (\frac {c_1}{2}\right )\right ) \sqrt {-8 i x+\cosh (c_1)+\sinh (c_1)} \\ y(x)\to -\frac {1}{4} \left (\cosh \left (\frac {c_1}{2}\right )+\sinh \left (\frac {c_1}{2}\right )\right ) \sqrt {8 i x+\cosh (c_1)+\sinh (c_1)} \\ y(x)\to \frac {1}{4} \left (\cosh \left (\frac {c_1}{2}\right )+\sinh \left (\frac {c_1}{2}\right )\right ) \sqrt {8 i x+\cosh (c_1)+\sinh (c_1)} \\ y(x)\to 0 \\ y(x)\to -x \\ y(x)\to x \\ \end{align*}