78.8.7 problem 7

Internal problem ID [18134]
Book : DIFFERENTIAL EQUATIONS WITH APPLICATIONS AND HISTORICAL NOTES by George F. Simmons. 3rd edition. 2017. CRC press, Boca Raton FL.
Section : Chapter 2. First order equations. Miscellaneous Problems for Chapter 2. Problems at page 99
Problem number : 7
Date solved : Monday, March 31, 2025 at 05:13:02 PM
CAS classification : [[_2nd_order, _missing_x], [_2nd_order, _exact, _nonlinear], _Liouville, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]]

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

✓ Maple. Time used: 0.022 (sec). Leaf size: 37

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

\begin{align*} y &= 0 \\ y &= \sqrt {c_1 \,{\mathrm e}^{2 x}+2 c_2} \\ y &= -\sqrt {c_1 \,{\mathrm e}^{2 x}+2 c_2} \\ \end{align*}

✓ Mathematica. Time used: 0.725 (sec). Leaf size: 38

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

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

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-2*y(x)*Derivative(y(x), x) + y(x)*Derivative(y(x), (x, 2)) + Derivative(y(x), x)**2,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

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