60.1.407 problem 409

Internal problem ID [10421]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, linear first order
Problem number : 409
Date solved : Monday, January 27, 2025 at 07:43:01 PM
CAS classification : [_rational, _dAlembert]

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

✓ Solution by Maple

Time used: 0.036 (sec). Leaf size: 65

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

\[ y = 2 x \,{\mathrm e}^{\operatorname {RootOf}\left (-x \,{\mathrm e}^{2 \textit {\_Z}}+2 x \,{\mathrm e}^{\textit {\_Z}}+2 \,{\mathrm e}^{\textit {\_Z}}+c_{1} -2 \textit {\_Z} -x \right )}-2 \operatorname {RootOf}\left (-x \,{\mathrm e}^{2 \textit {\_Z}}+2 x \,{\mathrm e}^{\textit {\_Z}}+2 \,{\mathrm e}^{\textit {\_Z}}+c_{1} -2 \textit {\_Z} -x \right )+c_{1} -x \]

✓ Solution by Mathematica

Time used: 1.378 (sec). Leaf size: 50

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

\[ \text {Solve}\left [\left \{x=\frac {2 K[1]-2 \log (K[1])}{(K[1]-1)^2}+\frac {c_1}{(K[1]-1)^2},y(x)=x K[1]^2-2 K[1]\right \},\{y(x),K[1]\}\right ] \]