9.19.16 problem 16

Internal problem ID [3300]
Book : Differential Equations by Alfred L. Nelson, Karl W. Folley, Max Coral. 3rd ed. DC heath. Boston. 1964
Section : Exercise 37, page 171
Problem number : 16
Date solved : Tuesday, September 30, 2025 at 06:33:00 AM
CAS classification : [[_homogeneous, `class A`], _rational, _dAlembert]

\begin{align*} \left ({y^{\prime }}^{2}-1\right ) x&=2 y^{\prime } y \end{align*}

✓ Maple. Time used: 0.027 (sec). Leaf size: 32

ode:=x*(diff(y(x),x)^2-1) = 2*y(x)*diff(y(x),x); 
dsolve(ode,y(x), singsol=all);
 

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

✓ Mathematica. Time used: 0.041 (sec). Leaf size: 45

ode=(D[y[x],x]^2-1)*x==2*D[y[x],x]*y[x]; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

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

✓ Sympy. Time used: 5.657 (sec). Leaf size: 22

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x*(Derivative(y(x), x)**2 - 1) - 2*y(x)*Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ \left [ y{\left (x \right )} = x \sinh {\left (C_{1} - \log {\left (x \right )} \right )}, \ y{\left (x \right )} = - x \sinh {\left (C_{1} - \log {\left (x \right )} \right )}\right ] \]