63.1.122 problem 181

Internal problem ID [15562]
Book : DIFFERENTIAL and INTEGRAL CALCULUS. VOL I. by N. PISKUNOV. MIR PUBLISHERS, Moscow 1969.
Section : Chapter 8. Differential equations. Exercises page 595
Problem number : 181
Date solved : Thursday, October 02, 2025 at 10:19:56 AM
CAS classification : [[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]]

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

✓ Maple. Time used: 0.043 (sec). Leaf size: 27

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

\begin{align*} y &= c_1 \cosh \left (\frac {c_2 +x}{c_1}\right ) \\ y &= c_1 \cosh \left (\frac {c_2 +x}{c_1}\right ) \\ \end{align*}

✓ Mathematica. Time used: 60.109 (sec). Leaf size: 128

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

\begin{align*} y(x)&\to -\frac {e^{-c_1} \tanh \left (\sqrt {e^{2 c_1}} (x+c_2)\right )}{\sqrt {-\text {sech}^2\left (\sqrt {e^{2 c_1}} (x+c_2)\right )}}\\ y(x)&\to \text {InverseFunction}\left [-\frac {\sqrt {1-\text {$\#$1}^2 e^{2 c_1}} \text {arcsinh}\left (\text {$\#$1} \sqrt {-e^{2 c_1}}\right )}{\sqrt {-e^{2 c_1}} \sqrt {-1+\text {$\#$1}^2 e^{2 c_1}}}\&\right ][x+c_2] \end{align*}

✗ Sympy

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

NotImplementedError : The given ODE -sqrt(y(x)*Derivative(y(x), (x, 2)) - 1) + Derivative(y(x), x) cannot be solved by the factorable group method