54.7.201 problem 1825 (book 6.234)

Internal problem ID [13050]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 6, non-linear second order
Problem number : 1825 (book 6.234)
Date solved : Wednesday, October 01, 2025 at 03:06:06 AM
CAS classification : [[_2nd_order, _missing_y]]

\begin{align*} \left (a \sqrt {{y^{\prime }}^{2}+1}-x y^{\prime }\right ) y^{\prime \prime }-{y^{\prime }}^{2}-1&=0 \end{align*}

✓ Maple. Time used: 0.705 (sec). Leaf size: 122

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

\begin{align*} y &= -i x +c_1 \\ y &= i x +c_1 \\ y &= \frac {c_2 a +\int \frac {-c_1 \,a^{2}+x \sqrt {a^{2} \left (c_1^{2}+a^{2}-x^{2}\right )}}{a^{2}-x^{2}}d x}{a} \\ y &= \frac {c_2 a -\int \frac {c_1 \,a^{2}+x \sqrt {a^{2} \left (c_1^{2}+a^{2}-x^{2}\right )}}{a^{2}-x^{2}}d x}{a} \\ \end{align*}

✓ Mathematica. Time used: 60.436 (sec). Leaf size: 331

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

\begin{align*} y(x)&\to -\frac {\sqrt {x^2 \left (a^2-x^2+c_1{}^2\right )} \left (c_1 \arctan \left (\frac {a^2-a x+c_1{}^2}{c_1 \sqrt {-a^2+x^2-c_1{}^2}}\right )+c_1 \arctan \left (\frac {a^2+a x+c_1{}^2}{c_1 \sqrt {-a^2+x^2-c_1{}^2}}\right )+2 \sqrt {-a^2+x^2-c_1{}^2}\right )}{2 x \sqrt {-a^2+x^2-c_1{}^2}}+c_1 \left (-\text {arctanh}\left (\frac {x}{a}\right )\right )+c_2\\ y(x)&\to \frac {\sqrt {x^2 \left (a^2-x^2+c_1{}^2\right )} \left (c_1 \arctan \left (\frac {a^2-a x+c_1{}^2}{c_1 \sqrt {-a^2+x^2-c_1{}^2}}\right )+c_1 \arctan \left (\frac {a^2+a x+c_1{}^2}{c_1 \sqrt {-a^2+x^2-c_1{}^2}}\right )+2 \sqrt {-a^2+x^2-c_1{}^2}\right )}{2 x \sqrt {-a^2+x^2-c_1{}^2}}+c_1 \left (-\text {arctanh}\left (\frac {x}{a}\right )\right )+c_2 \end{align*}

✗ Sympy

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

NotInvertible : zero divisor