78.8.11 problem 11

Internal problem ID [18138]
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 : 11
Date solved : Monday, March 31, 2025 at 05:13:20 PM
CAS classification : [[_2nd_order, _missing_y], [_2nd_order, _reducible, _mu_y_y1]]

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

✓ Maple. Time used: 0.023 (sec). Leaf size: 13

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

\[ y = -c_1 \,\operatorname {arctanh}\left (c_1 x \right )+c_2 \]

✓ Mathematica. Time used: 0.853 (sec). Leaf size: 36

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

\begin{align*} y(x)\to c_2-\frac {\text {arctanh}\left (\frac {x}{\sqrt {c_1}}\right )}{\sqrt {c_1}} \\ y(x)\to \text {Indeterminate} \\ y(x)\to c_2 \\ \end{align*}

✓ Sympy. Time used: 0.899 (sec). Leaf size: 51

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

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