29.25.34 problem 731

Internal problem ID [5321]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Various 25
Problem number : 731
Date solved : Sunday, March 30, 2025 at 07:55:17 AM
CAS classification : [[_homogeneous, `class G`], _dAlembert]

\begin{align*} x \left (x +\sqrt {x^{2}+y^{2}}\right ) y^{\prime }+y \sqrt {x^{2}+y^{2}}&=0 \end{align*}

✓ Maple. Time used: 0.022 (sec). Leaf size: 130

ode:=x*(x+(x^2+y(x)^2)^(1/2))*diff(y(x),x)+y(x)*(x^2+y(x)^2)^(1/2) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ -\int _{\textit {\_b}}^{x}\frac {\sqrt {\textit {\_a}^{2}+y^{2}}}{\textit {\_a} \left (2 \sqrt {\textit {\_a}^{2}+y^{2}}+\textit {\_a} \right )}d \textit {\_a} +\int _{}^{y}\frac {\textit {\_f}^{2} \left (2 \sqrt {\textit {\_f}^{2}+x^{2}}+x \right ) \int _{\textit {\_b}}^{x}\frac {1}{\sqrt {\textit {\_a}^{2}+\textit {\_f}^{2}}\, \left (2 \sqrt {\textit {\_a}^{2}+\textit {\_f}^{2}}+\textit {\_a} \right )^{2}}d \textit {\_a} -x -\sqrt {\textit {\_f}^{2}+x^{2}}}{\textit {\_f} \left (2 \sqrt {\textit {\_f}^{2}+x^{2}}+x \right )}d \textit {\_f} +c_1 = 0 \]

✓ Mathematica. Time used: 60.225 (sec). Leaf size: 1457

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

\begin{align*} \text {Solution too large to show}\end{align*}

✓ Sympy. Time used: 5.412 (sec). Leaf size: 48

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

\[ \log {\left (y{\left (x \right )} \right )} = C_{1} - \log {\left (\sqrt {\frac {x}{y{\left (x \right )}}} \sqrt [6]{\frac {x}{y{\left (x \right )}} + 2 \sqrt {\frac {x^{2}}{y^{2}{\left (x \right )}} + 1}} \right )} + \frac {\operatorname {asinh}{\left (\frac {x}{y{\left (x \right )}} \right )}}{3} \]