75.14.32 problem 358

Internal problem ID [16875]
Book : A book of problems in ordinary differential equations. M.L. KRASNOV, A.L. KISELYOV, G.I. MARKARENKO. MIR, MOSCOW. 1983
Section : Chapter 2 (Higher order ODEs). Section 14. Differential equations admitting of depression of their order. Exercises page 107
Problem number : 358
Date solved : Monday, March 31, 2025 at 03:34:23 PM
CAS classification : [[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]]

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

✓ Maple. Time used: 0.024 (sec). Leaf size: 22

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

\[ y = \frac {\left (c_{1}^{2}+1\right ) x^{2}}{4 c_{2}}+c_{1} x +c_{2} \]

✓ Mathematica. Time used: 0.029 (sec). Leaf size: 34

ode=2*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 {\left (1+c_1{}^2\right ) x^2}{4 c_2}+c_1 x+c_2 \\ y(x)\to \text {Indeterminate} \\ \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(2*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(2*y(x)*Derivative(y(x), (x, 2)) - 1) + Derivative(y(x), x) cannot be solved by the factorable group method