82.54.18 problem Ex. 18

Internal problem ID [18963]
Book : Introductory Course On Differential Equations by Daniel A Murray. Longmans Green and Co. NY. 1924
Section : Chapter IX. Equations of the second order. problems at end of chapter at page 120
Problem number : Ex. 18
Date solved : Monday, March 31, 2025 at 06:26:43 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*}

✓ Maple. Time used: 0.023 (sec). Leaf size: 39

ode:=y(x)*diff(diff(y(x),x),x)+diff(y(x),x)^2+1 = 0; 
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: 13.021 (sec). Leaf size: 83

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

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