23.4.65 problem 65

Internal problem ID [6367]
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 : 65
Date solved : Tuesday, September 30, 2025 at 02:53:23 PM
CAS classification : [[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]]

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

✓ Maple. Time used: 0.248 (sec). Leaf size: 104

ode:=diff(diff(y(x),x),x) = a*y(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 \\ a \int _{}^{y}\frac {\textit {\_a}^{2}+2 c_1}{\sqrt {4-\left (\textit {\_a}^{2}+2 c_1 \right )^{2} a^{2}}}d \textit {\_a} -x -c_2 &= 0 \\ -a \int _{}^{y}\frac {\textit {\_a}^{2}+2 c_1}{\sqrt {4-\left (\textit {\_a}^{2}+2 c_1 \right )^{2} a^{2}}}d \textit {\_a} -x -c_2 &= 0 \\ \end{align*}

✓ Mathematica. Time used: 1.19 (sec). Leaf size: 1104

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

Too large to display

✗ Sympy

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