81.2.3 problem 3

Internal problem ID [18540]
Book : A short course on differential equations. By Donald Francis Campbell. Maxmillan company. London. 1907
Section : Chapter II. Change of variable. Exercises at page 20
Problem number : 3
Date solved : Monday, March 31, 2025 at 05:41:46 PM
CAS classification : [[_2nd_order, _missing_x]]

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

✓ Maple. Time used: 0.055 (sec). Leaf size: 91

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

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

✓ Mathematica. Time used: 0.641 (sec). Leaf size: 71

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

\begin{align*} y(x)\to c_2-i \sqrt {\left (-1+c_1{}^2\right ) r^2+2 c_1 r x+x^2} \\ y(x)\to i \sqrt {\left (-1+c_1{}^2\right ) r^2+2 c_1 r x+x^2}+c_2 \\ \end{align*}

✗ Sympy

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

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