47.2.1 problem 8

Internal problem ID [9740]
Book : Elementary differential equations. By Earl D. Rainville, Phillip E. Bedient. Macmilliam Publishing Co. NY. 6th edition. 1981.
Section : CHAPTER 16. Nonlinear equations. Section 97. The p-discriminant equation. EXERCISES Page 314
Problem number : 8
Date solved : Tuesday, September 30, 2025 at 06:32:35 PM
CAS classification : [[_homogeneous, `class A`], _rational, _dAlembert]

\begin{align*} x {y^{\prime }}^{2}-2 y y^{\prime }+4 x&=0 \end{align*}

✓ Maple. Time used: 0.017 (sec). Leaf size: 30

ode:=x*diff(y(x),x)^2-2*y(x)*diff(y(x),x)+4*x = 0; 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= -2 x \\ y &= 2 x \\ y &= \frac {4 c_1^{2}+x^{2}}{2 c_1} \\ \end{align*}

✓ Mathematica. Time used: 0.165 (sec). Leaf size: 43

ode=x*(D[y[x],x])^2-2*y[x]*D[y[x],x]+4*x==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to -2 x \cosh (-\log (x)+c_1)\\ y(x)&\to -2 x \cosh (\log (x)+c_1)\\ y(x)&\to -2 x\\ y(x)&\to 2 x \end{align*}

✓ Sympy. Time used: 1.469 (sec). Leaf size: 15

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x*Derivative(y(x), x)**2 + 4*x - 2*y(x)*Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

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