53.4.27 problem 30

Internal problem ID [8515]
Book : Elementary differential equations. By Earl D. Rainville, Phillip E. Bedient. Macmilliam Publishing Co. NY. 6th edition. 1981.
Section : CHAPTER 16. Nonlinear equations. Section 101. Independent variable missing. EXERCISES Page 324
Problem number : 30
Date solved : Wednesday, March 05, 2025 at 06:01:59 AM
CAS classification : [[_2nd_order, _missing_x]]

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

✓ Maple. Time used: 0.059 (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 (x -1+c_{1} \right ) \sqrt {-\frac {1}{\left (c_{1} +x +1\right ) \left (x -1+c_{1} \right )}}+c_{2} \\ \end{align*}

✓ Mathematica. Time used: 0.254 (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