60.1.498 problem 511

Internal problem ID [10512]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, linear first order
Problem number : 511
Date solved : Sunday, March 30, 2025 at 05:35:52 PM
CAS classification : [[_1st_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 4.224 (sec). Leaf size: 309

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

\begin{align*} y &= -i x \\ y &= i x \\ \frac {2 \sqrt {a^{2} \left (y^{2}+x^{2}\right )^{2} \left (-a^{2}+\sqrt {y^{2}+x^{2}}\right )}\, \sqrt {-a^{2}+\sqrt {y^{2}+x^{2}}}\, \arctan \left (\frac {\sqrt {-a^{2}+\sqrt {y^{2}+x^{2}}}}{a}\right )-a \left (y^{2}+x^{2}\right ) \left (a^{2}-\sqrt {y^{2}+x^{2}}\right ) \left (c_1 -\arctan \left (\frac {x}{y}\right )\right )}{a \left (y^{2}+x^{2}\right ) \left (a^{2}-\sqrt {y^{2}+x^{2}}\right )} &= 0 \\ \frac {-2 \sqrt {a^{2} \left (y^{2}+x^{2}\right )^{2} \left (-a^{2}+\sqrt {y^{2}+x^{2}}\right )}\, \sqrt {-a^{2}+\sqrt {y^{2}+x^{2}}}\, \arctan \left (\frac {\sqrt {-a^{2}+\sqrt {y^{2}+x^{2}}}}{a}\right )-a \left (y^{2}+x^{2}\right ) \left (a^{2}-\sqrt {y^{2}+x^{2}}\right ) \left (c_1 -\arctan \left (\frac {x}{y}\right )\right )}{a \left (y^{2}+x^{2}\right ) \left (a^{2}-\sqrt {y^{2}+x^{2}}\right )} &= 0 \\ \end{align*}

✓ Mathematica. Time used: 107.431 (sec). Leaf size: 3159

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

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✗ Sympy

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

Timed Out