73.8.24 problem 13.4 (e)

Internal problem ID [15161]
Book : Ordinary Differential Equations. An introduction to the fundamentals. Kenneth B. Howell. second edition. CRC Press. FL, USA. 2020
Section : Chapter 13. Higher order equations: Extending first order concepts. Additional exercises page 259
Problem number : 13.4 (e)
Date solved : Monday, March 31, 2025 at 01:29:40 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^{\prime }}^{2}+y y^{\prime \prime }&=2 y y^{\prime } \end{align*}

✓ Maple. Time used: 0.021 (sec). Leaf size: 37

ode:=diff(y(x),x)^2+y(x)*diff(diff(y(x),x),x) = 2*y(x)*diff(y(x),x); 
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: 0.484 (sec). Leaf size: 45

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

\[ y(x)\to c_2 \exp \left (\int _1^x\text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {1}{(K[1]-1) K[1]}dK[1]\&\right ][c_1-2 K[2]]dK[2]\right ) \]

✗ 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