52.1.8 problem 8

Internal problem ID [10379]
Book : Second order enumerated odes
Section : section 1
Problem number : 8
Date solved : Tuesday, September 30, 2025 at 07:22:47 PM
CAS classification : [[_2nd_order, _quadrature]]

\begin{align*} {y^{\prime \prime }}^{2}&=1 \end{align*}

✓ Maple. Time used: 0.002 (sec). Leaf size: 27

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

\begin{align*} y &= \frac {1}{2} x^{2}+c_1 x +c_2 \\ y &= -\frac {1}{2} x^{2}+c_1 x +c_2 \\ \end{align*}

✓ Mathematica. Time used: 0.002 (sec). Leaf size: 37

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

\begin{align*} y(x)&\to -\frac {x^2}{2}+c_2 x+c_1\\ y(x)&\to \frac {x^2}{2}+c_2 x+c_1 \end{align*}

✓ Sympy. Time used: 0.095 (sec). Leaf size: 24

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

\[ \left [ y{\left (x \right )} = C_{1} + C_{2} x - \frac {x^{2}}{2}, \ y{\left (x \right )} = C_{1} + C_{2} x + \frac {x^{2}}{2}\right ] \]