75.13.8 problem 325

Internal problem ID [16842]
Book : A book of problems in ordinary differential equations. M.L. KRASNOV, A.L. KISELYOV, G.I. MARKARENKO. MIR, MOSCOW. 1983
Section : Chapter 2 (Higher order ODEs). Section 13. Basic concepts and definitions. Exercises page 98
Problem number : 325
Date solved : Monday, March 31, 2025 at 03:24:01 PM
CAS classification : [[_2nd_order, _missing_x]]

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

✓ Maple. Time used: 0.067 (sec). Leaf size: 47

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

✓ Mathematica. Time used: 0.239 (sec). Leaf size: 59

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

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

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-(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)**(1/3)/2 + sqrt(3)*I*(Derivative(y(x), (x, 2))**2)**(1/3)/2 - 1) + Derivative(y(x), x) cannot be solved by the factorable group method