60.7.48 problem 1652 (book 6.61)

Internal problem ID [11598]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 6, non-linear second order
Problem number : 1652 (book 6.61)
Date solved : Sunday, March 30, 2025 at 08:30:47 PM
CAS classification : [[_2nd_order, _missing_x]]

\begin{align*} y^{\prime \prime }&=a \sqrt {1+{y^{\prime }}^{2}}+b \end{align*}

✓ Maple. Time used: 0.053 (sec). Leaf size: 31

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

\[ y = \int \operatorname {RootOf}\left (x -\int _{}^{\textit {\_Z}}\frac {1}{a \sqrt {\textit {\_f}^{2}+1}+b}d \textit {\_f} +c_1 \right )d x +c_2 \]

✓ Mathematica. Time used: 1.884 (sec). Leaf size: 46

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

\[ y(x)\to \int _1^x\text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {1}{\sqrt {K[1]^2+1} a+b}dK[1]\&\right ][c_1+K[2]]dK[2]+c_2 \]

✗ Sympy

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

NotImplementedError : The given ODE -sqrt(-1 + (b - Derivative(y(x), (x, 2)))**2/a**2) + Derivative(y(x), x) cannot be solved by the factorable group method