75.13.9 problem 326

Internal problem ID [16843]
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 13. Basic concepts and definitions. Exercises page 98
Problem number : 326
Date solved : Monday, March 31, 2025 at 03:24:04 PM
CAS classification : [[_2nd_order, _missing_x], [_2nd_order, _exact, _nonlinear], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]]

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

✓ Maple. Time used: 0.024 (sec). Leaf size: 35

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

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

✓ Mathematica. Time used: 0.597 (sec). Leaf size: 79

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

\begin{align*} y(x)\to -\sqrt {(x+c_2){}^2-e^{2 c_1}} \\ y(x)\to \sqrt {(x+c_2){}^2-e^{2 c_1}} \\ y(x)\to -\sqrt {(x+c_2){}^2} \\ y(x)\to \sqrt {(x+c_2){}^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