75.22.5 problem 710

Internal problem ID [17113]
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 17. Boundary value problems. Exercises page 163
Problem number : 710
Date solved : Monday, March 31, 2025 at 03:42:11 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 y^{\prime \prime }+{y^{\prime }}^{2}+1&=0 \end{align*}

With initial conditions

\begin{align*} y \left (0\right )&=1\\ y \left (1\right )&=2 \end{align*}

✓ Maple. Time used: 0.725 (sec). Leaf size: 16

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

\[ y = \sqrt {-x^{2}+4 x +1} \]

✓ Mathematica. Time used: 11.627 (sec). Leaf size: 19

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

\[ y(x)\to \sqrt {-x^2+4 x+1} \]

✗ 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 = {y(0): 1, y(1): 2} 
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