29.27.23 problem 789

Internal problem ID [5373]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Various 27
Problem number : 789
Date solved : Monday, January 27, 2025 at 11:17:28 AM
CAS classification : [_quadrature]

\begin{align*} {y^{\prime }}^{2}-2 x y^{\prime }+1&=0 \end{align*}

✓ Solution by Maple

Time used: 0.033 (sec). Leaf size: 65

dsolve(diff(y(x),x)^2-2*x*diff(y(x),x)+1 = 0,y(x), singsol=all)
 

\begin{align*} y \left (x \right ) &= \frac {x^{2}}{2}-\frac {\sqrt {x^{2}-1}\, x}{2}+\frac {\ln \left (x +\sqrt {x^{2}-1}\right )}{2}+c_{1} \\ y \left (x \right ) &= \frac {x^{2}}{2}+\frac {\sqrt {x^{2}-1}\, x}{2}-\frac {\ln \left (x +\sqrt {x^{2}-1}\right )}{2}+c_{1} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.015 (sec). Leaf size: 82

DSolve[(D[y[x],x])^2-2*x*D[y[x],x]+1==0,y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to \frac {1}{2} \left (-\text {arctanh}\left (\frac {x}{\sqrt {x^2-1}}\right )+x^2+\sqrt {x^2-1} x+2 c_1\right ) \\ y(x)\to \frac {1}{2} \left (\text {arctanh}\left (\frac {x}{\sqrt {x^2-1}}\right )+x^2-\sqrt {x^2-1} x+2 c_1\right ) \\ \end{align*}