54.1.544 problem 557

Internal problem ID [11858]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, linear first order
Problem number : 557
Date solved : Tuesday, September 30, 2025 at 11:29:41 PM
CAS classification : [[_homogeneous, `class A`], _rational, _Bernoulli]

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

✓ Maple. Time used: 0.101 (sec). Leaf size: 97

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

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

✓ Mathematica. Time used: 0.186 (sec). Leaf size: 37

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

\begin{align*} y(x)&\to -\sqrt {-x (x-c_1)}\\ y(x)&\to \sqrt {-x (x-c_1)} \end{align*}

✓ Sympy. Time used: 0.312 (sec). Leaf size: 22

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

\[ \left [ y{\left (x \right )} = - \sqrt {x \left (C_{1} - x\right )}, \ y{\left (x \right )} = \sqrt {x \left (C_{1} - x\right )}\right ] \]