54.1.330 problem 337

Internal problem ID [11644]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, linear first order
Problem number : 337
Date solved : Tuesday, September 30, 2025 at 09:53:08 PM
CAS classification : [[_homogeneous, `class A`], _rational, _dAlembert]

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

✓ Maple. Time used: 0.090 (sec). Leaf size: 27

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

\[ \frac {-c_1 y^{2}+\sqrt {y^{2}+x^{2}}+x}{y^{2}} = 0 \]

✓ Mathematica. Time used: 0.277 (sec). Leaf size: 57

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

\begin{align*} y(x)&\to -e^{\frac {c_1}{2}} \sqrt {2 x+e^{c_1}}\\ y(x)&\to e^{\frac {c_1}{2}} \sqrt {2 x+e^{c_1}}\\ y(x)&\to 0 \end{align*}

✓ Sympy. Time used: 1.353 (sec). Leaf size: 12

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

\[ \log {\left (y{\left (x \right )} \right )} = C_{1} + \operatorname {asinh}{\left (\frac {x}{y{\left (x \right )}} \right )} \]