63.1.80 problem 125

Internal problem ID [15520]
Book : DIFFERENTIAL and INTEGRAL CALCULUS. VOL I. by N. PISKUNOV. MIR PUBLISHERS, Moscow 1969.
Section : Chapter 8. Differential equations. Exercises page 595
Problem number : 125
Date solved : Thursday, October 02, 2025 at 10:19:24 AM
CAS classification : [[_2nd_order, _missing_x]]

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

With initial conditions

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

✓ Maple. Time used: 0.293 (sec). Leaf size: 24

ode:=diff(diff(y(x),x),x)^2+diff(y(x),x)^2 = a^2; 
ic:=[y(0) = -1, D(y)(0) = 0]; 
dsolve([ode,op(ic)],y(x), singsol=all);
 

\begin{align*} y &= -1-a +a \cos \left (x \right ) \\ y &= a -1-a \cos \left (x \right ) \\ \end{align*}

✓ Mathematica. Time used: 4.722 (sec). Leaf size: 37

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

\begin{align*} y(x)&\to a \left (\frac {1}{\sqrt {\sec ^2(x)}}-1\right )-1\\ y(x)&\to -\frac {a}{\sqrt {\sec ^2(x)}}+a-1 \end{align*}

✗ Sympy

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

NotImplementedError : The given ODE -sqrt(a**2 - Derivative(y(x), (x, 2))**2) + Derivative(y(x), x) cannot be solved by the factorable group method