23.2.358 problem 410

Internal problem ID [5713]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Part II. Chapter 2. THE DIFFERENTIAL EQUATION IS OF FIRST ORDER AND OF SECOND OR HIGHER DEGREE, page 278
Problem number : 410
Date solved : Tuesday, September 30, 2025 at 02:01:49 PM
CAS classification : [_Clairaut]

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

✓ Maple. Time used: 0.110 (sec). Leaf size: 37

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

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

✓ Mathematica. Time used: 0.613 (sec). Leaf size: 43

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

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

✗ Sympy

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

NotImplementedError : multiple generators [_X0, log(_X0 + sqrt(_X0**2 + 1)), sqrt(_X0**2 + 1)] 
No al