15.20.4 problem 4

Internal problem ID [3312]
Book : Differential Equations by Alfred L. Nelson, Karl W. Folley, Max Coral. 3rd ed. DC heath. Boston. 1964
Section : Exercise 38, page 173
Problem number : 4
Date solved : Tuesday, March 04, 2025 at 04:34:22 PM
CAS classification : [[_homogeneous, `class A`], _rational, _dAlembert]

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

✓ Maple. Time used: 0.010 (sec). Leaf size: 30

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

\begin{align*} y \left (x \right ) &= -2 x \\ y \left (x \right ) &= 2 x \\ y \left (x \right ) &= \frac {4 c_{1}^{2}+x^{2}}{2 c_{1}} \\ \end{align*}

✓ Mathematica. Time used: 0.26 (sec). Leaf size: 43

ode=4*x-2*D[y[x],x]*y[x]+x*D[y[x],x]^2==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: 2.126 (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)
 

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