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 : Sunday, March 30, 2025 at 08:03:51 AM
CAS classification : [_quadrature]

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

✓ Maple. Time used: 0.028 (sec). Leaf size: 65

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

\begin{align*} y &= \frac {x^{2}}{2}-\frac {x \sqrt {x^{2}-1}}{2}+\frac {\ln \left (x +\sqrt {x^{2}-1}\right )}{2}+c_1 \\ y &= \frac {x^{2}}{2}+\frac {x \sqrt {x^{2}-1}}{2}-\frac {\ln \left (x +\sqrt {x^{2}-1}\right )}{2}+c_1 \\ \end{align*}

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

ode=(D[y[x],x])^2-2*x*D[y[x],x]+1==0; 
ic={}; 
DSolve[{ode,ic},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*}

✓ Sympy. Time used: 0.311 (sec). Leaf size: 68

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-2*x*Derivative(y(x), x) + Derivative(y(x), x)**2 + 1,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ \left [ y{\left (x \right )} = C_{1} + \frac {x^{2}}{2} - \frac {x \sqrt {x^{2} - 1}}{2} + \frac {\log {\left (x + \sqrt {x^{2} - 1} \right )}}{2}, \ y{\left (x \right )} = C_{1} + \frac {x^{2}}{2} + \frac {x \sqrt {x^{2} - 1}}{2} - \frac {\log {\left (x + \sqrt {x^{2} - 1} \right )}}{2}\right ] \]