75.1.5 problem 4

Internal problem ID [19886]
Book : A short course on differential equations. By Donald Francis Campbell. Maxmillan company. London. 1907
Section : Chapter I. Introduction. Exercises at page 13
Problem number : 4
Date solved : Thursday, October 02, 2025 at 05:00:17 PM
CAS classification : [[_2nd_order, _missing_x]]

\begin{align*} 1+{y^{\prime }}^{2}+\frac {m y^{\prime \prime }}{\sqrt {1+{y^{\prime }}^{2}}}&=0 \end{align*}

✓ Maple. Time used: 0.044 (sec). Leaf size: 75

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

\begin{align*} y &= \frac {\left (m +x +c_1 \right ) \left (-m +x +c_1 \right )}{\sqrt {-c_1^{2}-2 c_1 x +m^{2}-x^{2}}}+c_2 \\ y &= \frac {\left (m +x +c_1 \right ) \left (m -x -c_1 \right )}{\sqrt {-c_1^{2}-2 c_1 x +m^{2}-x^{2}}}+c_2 \\ \end{align*}

✓ Mathematica. Time used: 0.531 (sec). Leaf size: 71

ode=1+D[y[x],x]^2+m/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 c_2-i \sqrt {\left (-1+c_1{}^2\right ) m^2-2 c_1 m x+x^2}\\ y(x)&\to i \sqrt {\left (-1+c_1{}^2\right ) m^2-2 c_1 m x+x^2}+c_2 \end{align*}

✗ Sympy

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

NotImplementedError : The given ODE -sqrt(-(m**2*Derivative(y(x), (x, 2))**2)**(1/3)/2 + sqrt(3)*I*(m**2*Derivative(y(x), (x, 2))**2)**(1/3)/2 - 1) + Derivative(y(x), x) cannot be solved by the factorable group method