23.4.63 problem 63

Internal problem ID [6365]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Part II. Chapter 4. THE NONLINEAR EQUATION OF SECOND ORDER, page 380
Problem number : 63
Date solved : Tuesday, September 30, 2025 at 02:53:18 PM
CAS classification : [[_2nd_order, _missing_x]]

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

✓ Maple. Time used: 0.171 (sec). Leaf size: 59

ode:=diff(diff(y(x),x),x) = a*(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+a^{2} \left (c_1 +x \right )^{2}\right ) \sqrt {-\frac {1}{-1+a^{2} \left (c_1 +x \right )^{2}}}+c_2 a}{a} \\ \end{align*}

✓ Mathematica. Time used: 0.342 (sec). Leaf size: 75

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

✗ Sympy

from sympy import * 
x = symbols("x") 
a = symbols("a") 
y = Function("y") 
ode = Eq(-a*(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/a**2)**(1/3)/2 + sqrt(3)*I*(