47.4.25 problem 27

Internal problem ID [9799]
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 : 27
Date solved : Tuesday, September 30, 2025 at 06:41:54 PM
CAS classification : [[_2nd_order, _missing_y], [_2nd_order, _reducible, _mu_y_y1]]

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

✓ Maple. Time used: 0.020 (sec). Leaf size: 21

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

\[ y = \frac {x}{c_1}+\frac {\ln \left (c_1 x -1\right )}{c_1^{2}}+c_2 \]

✓ Mathematica. Time used: 0.441 (sec). Leaf size: 51

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

\begin{align*} y(x)&\to \int _1^x-\frac {K[1]}{c_1 K[1]+1}dK[1]+c_2\\ y(x)&\to c_2\\ y(x)&\to -\frac {x^2}{2}+\frac {1}{2}+c_2 \end{align*}

✓ Sympy. Time used: 0.391 (sec). Leaf size: 17

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

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