82.8.3 problem 36-4

Internal problem ID [21874]
Book : The Differential Equations Problem Solver. VOL. II. M. Fogiel director. REA, NY. 1978. ISBN 78-63609
Section : Chapter 36. Nonlinear differential equations. Page 1203
Problem number : 36-4
Date solved : Thursday, October 02, 2025 at 08:03:13 PM
CAS classification : [[_2nd_order, _quadrature]]

\begin{align*} {y^{\prime \prime }}^{2} x^{2} \left (x^{2}-1\right )-1&=0 \end{align*}

✓ Maple. Time used: 0.019 (sec). Leaf size: 60

ode:=diff(diff(y(x),x),x)^2*x^2*(x^2-1)-1 = 0; 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= -x \arctan \left (\frac {1}{\sqrt {x^{2}-1}}\right )-\ln \left (x +\sqrt {x^{2}-1}\right )+c_1 x +c_2 \\ y &= x \arctan \left (\frac {1}{\sqrt {x^{2}-1}}\right )+\ln \left (x +\sqrt {x^{2}-1}\right )+c_1 x +c_2 \\ \end{align*}

✓ Mathematica. Time used: 0.017 (sec). Leaf size: 126

ode=D[y[x],{x,2}]^2*x^2*(x^2-1)-1==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to -\frac {x \sqrt {x^2-1} \left (x \arctan \left (\sqrt {x^2-1}\right )-\log \left (\sqrt {x^2-1}+x\right )\right )}{\sqrt {x^2 \left (x^2-1\right )}}+c_2 x+c_1\\ y(x)&\to \frac {x \sqrt {x^2-1} \left (x \arctan \left (\sqrt {x^2-1}\right )-\log \left (\sqrt {x^2-1}+x\right )\right )}{\sqrt {x^2 \left (x^2-1\right )}}+c_2 x+c_1 \end{align*}

✓ Sympy. Time used: 4.640 (sec). Leaf size: 92

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

\[ \left [ y{\left (x \right )} = \begin {cases} C x + C - x \operatorname {acos}{\left (\frac {1}{x} \right )} + \operatorname {acosh}{\left (x \right )} & \text {for}\: x > -1 \wedge \left |{x^{2}}\right | > 1 \wedge x < 1 \\C x + C - x \operatorname {acos}{\left (\frac {1}{x} \right )} - i \operatorname {asin}{\left (x \right )} & \text {for}\: x > -1 \wedge x < 1 \end {cases}, \ y{\left (x \right )} = \begin {cases} C x + C + x \operatorname {acos}{\left (\frac {1}{x} \right )} - \operatorname {acosh}{\left (x \right )} & \text {for}\: x > -1 \wedge \left |{x^{2}}\right | > 1 \wedge x < 1 \\C x + C + x \operatorname {acos}{\left (\frac {1}{x} \right )} + i \operatorname {asin}{\left (x \right )} & \text {for}\: x > -1 \wedge x < 1 \end {cases}\right ] \]