Internal problem ID [10200]
Book: An elementary treatise on differential equations by Abraham Cohen. DC heath publishers.
1906
Section: Chapter IV, differential equations of the first order and higher degree than the first. Article
26. Equations solvable for \(x\). Page 55
Problem number: Ex 3.
ODE order: 1.
ODE degree: 2.
CAS Maple gives this as type [[_homogeneous, class A], _rational, _dAlembert]
Solve \begin {gather*} \boxed {x \left (y^{\prime }\right )^{2}-2 y^{\prime } y-x=0} \end {gather*}
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 23
dsolve(x*diff(y(x),x)^2-2*y(x)*diff(y(x),x)-x=0,y(x), singsol=all)
\begin{align*} y \relax (x ) = -\frac {\left (-\frac {x^{2}}{c_{1}^{2}}+1\right ) c_{1}}{2} \\ y \relax (x ) = x c_{1} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.129 (sec). Leaf size: 71
DSolve[x*(y'[x])^2-2*y[x]*y'[x]-x==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {1}{2} e^{-c_1} \left (-x^2+e^{2 c_1}\right ) \\ y(x)\to \frac {1}{2} e^{-c_1} \left (-1+e^{2 c_1} x^2\right ) \\ y(x)\to -i x \\ y(x)\to i x \\ \end{align*}