15.8.29 problem 30

Internal problem ID [3032]
Book : Differential Equations by Alfred L. Nelson, Karl W. Folley, Max Coral. 3rd ed. DC heath. Boston. 1964
Section : Exercise 12, page 46
Problem number : 30
Date solved : Tuesday, March 04, 2025 at 03:45:56 PM
CAS classification : [[_homogeneous, `class A`], _dAlembert]

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

✓ Maple. Time used: 0.082 (sec). Leaf size: 29

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

\[ \frac {-c_{1} y+\sqrt {x^{2}+y^{2}}\, x +x^{2}}{y} = 0 \]

✓ Mathematica. Time used: 0.325 (sec). Leaf size: 47

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

\begin{align*} y(x)\to -x \sqrt {-\text {sech}^2(\log (x)+c_1)} \\ y(x)\to x \sqrt {-\text {sech}^2(\log (x)+c_1)} \\ y(x)\to 0 \\ \end{align*}

✓ Sympy. Time used: 2.054 (sec). Leaf size: 10

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

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