53.4.22 problem 24

Internal problem ID [8510]
Book : Elementary differential equations. By Earl D. Rainville, Phillip E. Bedient. Macmilliam Publishing Co. NY. 6th edition. 1981.
Section : CHAPTER 16. Nonlinear equations. Section 101. Independent variable missing. EXERCISES Page 324
Problem number : 24
Date solved : Sunday, March 30, 2025 at 01:12:45 PM
CAS classification : [[_2nd_order, _missing_x], _Liouville, [_2nd_order, _reducible, _mu_xy]]

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

✓ Maple. Time used: 0.019 (sec). Leaf size: 15

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

\[ y = -\ln \left (-c_1 x -c_2 \right ) \]

✓ Mathematica. Time used: 0.214 (sec). Leaf size: 15

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

\[ y(x)\to c_2-\log (x+c_1) \]

✓ Sympy. Time used: 0.470 (sec). Leaf size: 8

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

\[ y{\left (x \right )} = C_{1} - \log {\left (C_{2} + x \right )} \]