78.8.46 problem 46

Internal problem ID [18173]
Book : DIFFERENTIAL EQUATIONS WITH APPLICATIONS AND HISTORICAL NOTES by George F. Simmons. 3rd edition. 2017. CRC press, Boca Raton FL.
Section : Chapter 2. First order equations. Miscellaneous Problems for Chapter 2. Problems at page 99
Problem number : 46
Date solved : Monday, March 31, 2025 at 05:21:46 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: 0.022 (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: 0.515 (sec). Leaf size: 47

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 -\frac {x}{c_1}+\frac {\log (1+c_1 x)}{c_1{}^2}+c_2 \\ y(x)\to c_2 \\ y(x)\to -\frac {x^2}{2}+c_2 \\ \end{align*}

✓ Sympy. Time used: 0.614 (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)
 

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