44.7.4 problem 1(d)

Internal problem ID [9194]
Book : Differential Equations: Theory, Technique, and Practice by George Simmons, Steven Krantz. McGraw-Hill NY. 2007. 1st Edition.
Section : Chapter 1. What is a differential equation. Section 1.9. Reduction of Order. Page 38
Problem number : 1(d)
Date solved : Tuesday, September 30, 2025 at 06:12:30 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: 0.023 (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: 10.911 (sec). Leaf size: 52

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]
 

\begin{align*} y(x)&\to \int _1^x\frac {K[1]^2}{c_1-K[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: 0.597 (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)
 

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