7.5.49 problem 49

Internal problem ID [153]
Book : Elementary Differential Equations. By C. Henry Edwards, David E. Penney and David Calvis. 6th edition. 2008
Section : Chapter 1. First order differential equations. Section 1.6 (substitution and exact equations). Problems at page 72
Problem number : 49
Date solved : Saturday, March 29, 2025 at 04:37:13 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}&=y y^{\prime } \end{align*}

✓ Maple. Time used: 0.027 (sec). Leaf size: 35

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

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

✓ Mathematica. Time used: 0.778 (sec). Leaf size: 41

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

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

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-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) + 4*Derivative(y(x), (x, 2)))*y(x))/2 - y(x)/2 + Derivative(y(x), x) cannot be solved by the factorable group method