75.1.2 problem 1 (b)

Internal problem ID [19883]
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 : 1 (b)
Date solved : Thursday, October 02, 2025 at 05:00:14 PM
CAS classification : [[_2nd_order, _missing_x]]

\begin{align*} y^{\prime \prime }&=c \left (1+{y^{\prime }}^{2}\right )^{{3}/{2}} \end{align*}

✓ Maple. Time used: 0.252 (sec). Leaf size: 57

ode:=diff(diff(y(x),x),x) = c*(1+diff(y(x),x)^2)^(3/2); 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= -i x +c_1 \\ y &= i x +c_1 \\ y &= \frac {\left (-1+c^{2} \left (c_1 +x \right )^{2}\right ) \sqrt {-\frac {1}{-1+c^{2} \left (c_1 +x \right )^{2}}}+c_2 c}{c} \\ \end{align*}

✓ Mathematica. Time used: 0.49 (sec). Leaf size: 75

ode=D[y[x],{x,2}]==c*(1+D[y[x],x]^2)^(3/2); 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to c_2-\frac {i \sqrt {c^2 x^2+2 c c_1 x-1+c_1{}^2}}{c}\\ y(x)&\to \frac {i \sqrt {c^2 x^2+2 c c_1 x-1+c_1{}^2}}{c}+c_2 \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
c = symbols("c") 
y = Function("y") 
ode = Eq(-c*(Derivative(y(x), x)**2 + 1)**(3/2) + Derivative(y(x), (x, 2)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

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