83.31.2 problem 2

Internal problem ID [19315]
Book : A Text book for differentional equations for postgraduate students by Ray and Chaturvedi. First edition, 1958. BHASKAR press. INDIA
Section : Chapter VII. Exact differential equations and certain particular forms of equations. Exercise VII (E) at page 112
Problem number : 2
Date solved : Thursday, March 13, 2025 at 02:13:37 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 \end{align*}

✓ Maple. Time used: 0.040 (sec). Leaf size: 35

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

\begin{align*} y \left (x \right ) &= \sqrt {-2 c_{1} x +x^{2}+2 c_{2}} \\ y \left (x \right ) &= -\sqrt {-2 c_{1} x +x^{2}+2 c_{2}} \\ \end{align*}

✓ Mathematica. Time used: 0.592 (sec). Leaf size: 79

ode=y[x]*D[y[x],{x,2}]+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