62.18.2 problem Ex 2

Internal problem ID [12913]
Book : An elementary treatise on differential equations by Abraham Cohen. DC heath publishers. 1906
Section : Chapter V, Singular solutions. Article 30. Page 63
Problem number : Ex 2
Date solved : Tuesday, January 28, 2025 at 04:43:20 AM
CAS classification : [[_homogeneous, `class A`], _rational, _dAlembert]

\begin{align*} x {y^{\prime }}^{2}-2 y^{\prime } y-x&=0 \end{align*}

✓ Solution by Maple

Time used: 0.064 (sec). Leaf size: 30

dsolve(x*diff(y(x),x)^2-2*y(x)*diff(y(x),x)-x=0,y(x), singsol=all)
 

\begin{align*} y &= -i x \\ y &= i x \\ y &= \frac {-c_{1}^{2}+x^{2}}{2 c_{1}} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.088 (sec). Leaf size: 45

DSolve[x*(D[y[x],x])^2-2*y[x]*D[y[x],x]-x==0,y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to x \sinh (-\log (x)+c_1) \\ y(x)\to x \sinh (\log (x)+c_1) \\ y(x)\to -i x \\ y(x)\to i x \\ \end{align*}