78.7.4 problem 1 (d)

Internal problem ID [18119]
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. Section 11 (Reduction of order). Problems at page 87
Problem number : 1 (d)
Date solved : Monday, March 31, 2025 at 05:12:15 PM
CAS classification : [[_2nd_order, _missing_y]]

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

✓ Maple. Time used: 0.028 (sec). Leaf size: 26

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

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

✓ Mathematica. Time used: 0.4 (sec). Leaf size: 41

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

\[ y(x)\to -\frac {x^2}{2}-c_1 x-c_1{}^2 \log (x-c_1)+\frac {3 c_1{}^2}{2}+c_2 \]

✓ Sympy. Time used: 0.933 (sec). Leaf size: 20

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

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