83.31.3 problem 3

Internal problem ID [19316]
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 : 3
Date solved : Thursday, March 13, 2025 at 02:13:41 PM
CAS classification : [[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]]

\begin{align*} y y^{\prime \prime }-{y^{\prime }}^{2}+y^{\prime }&=0 \end{align*}

✓ Maple. Time used: 0.053 (sec). Leaf size: 20

ode:=y(x)*diff(diff(y(x),x),x)-diff(y(x),x)^2+diff(y(x),x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y \left (x \right ) &= 0 \\ y \left (x \right ) &= \frac {-1+{\mathrm e}^{\left (c_{2} +x \right ) c_{1}}}{c_{1}} \\ \end{align*}

✓ Mathematica. Time used: 1.495 (sec). Leaf size: 29

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

\begin{align*} y(x)\to \frac {-1+e^{c_1 (x+c_2)}}{c_1} \\ y(x)\to x+c_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 + Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE -sqrt(4*y(x)*Derivative(y(x), (x, 2)) + 1)/2 + Derivative(y(x), x) - 1/2 cannot be solved by the factorable group method