73.8.53 problem 13.9 (iv)

Internal problem ID [15182]
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 : 13.9 (iv)
Date solved : Thursday, March 13, 2025 at 05:48:55 AM
CAS classification : [[_2nd_order, _missing_x], [_2nd_order, _exact, _nonlinear], _Lagerstrom, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]]

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

With initial conditions

\begin{align*} y \left (0\right )&=0\\ y^{\prime }\left (0\right )&=-1 \end{align*}

✓ Maple. Time used: 0.132 (sec). Leaf size: 8

ode:=diff(diff(y(x),x),x) = 2*y(x)*diff(y(x),x); 
ic:=y(0) = 0, D(y)(0) = -1; 
dsolve([ode,ic],y(x), singsol=all);
 

\[ y = -\tanh \left (x \right ) \]

✓ Mathematica. Time used: 1.687 (sec). Leaf size: 9

ode=D[y[x],{x,2}]==2*y[x]*D[y[x],x]; 
ic={y[0]==0,Derivative[1][y][0] ==-1}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to -\tanh (x) \]

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-2*y(x)*Derivative(y(x), x) + Derivative(y(x), (x, 2)),0) 
ics = {y(0): 0, Subs(Derivative(y(x), x), x, 0): -1} 
dsolve(ode,func=y(x),ics=ics)
 

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