54.1.503 problem 516

Internal problem ID [11817]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, linear first order
Problem number : 516
Date solved : Friday, October 03, 2025 at 02:46:29 AM
CAS classification : [[_homogeneous, `class A`]]

\begin{align*} \left (y^{2}+x^{2}\right ) f \left (\frac {x}{\sqrt {y^{2}+x^{2}}}\right ) \left ({y^{\prime }}^{2}+1\right )-\left (x y^{\prime }-y\right )^{2}&=0 \end{align*}

✓ Maple. Time used: 0.631 (sec). Leaf size: 70

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

\[ y = \operatorname {RootOf}\left (-\ln \left (x \right )-\int _{}^{\textit {\_Z}}\frac {\textit {\_a} f \left (\frac {1}{\sqrt {\textit {\_a}^{2}+1}}\right )+\sqrt {-f \left (\frac {1}{\sqrt {\textit {\_a}^{2}+1}}\right ) \left (f \left (\frac {1}{\sqrt {\textit {\_a}^{2}+1}}\right )-1\right )}}{\left (\textit {\_a}^{2}+1\right ) f \left (\frac {1}{\sqrt {\textit {\_a}^{2}+1}}\right )}d \textit {\_a} +c_1 \right ) x \]

✓ Mathematica. Time used: 0.937 (sec). Leaf size: 253

ode=-(-y[x] + x*D[y[x],x])^2 + f[x/Sqrt[x^2 + y[x]^2]]*(x^2 + y[x]^2)*(1 + D[y[x],x]^2)==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} \text {Solve}\left [\int _1^{\frac {y(x)}{x}}\frac {f\left (\frac {1}{\sqrt {K[1]^2+1}}\right ) K[1]^2+f\left (\frac {1}{\sqrt {K[1]^2+1}}\right )-1}{\sqrt {f\left (\frac {1}{\sqrt {K[1]^2+1}}\right )} (K[1]-i) (K[1]+i) \left (\sqrt {f\left (\frac {1}{\sqrt {K[1]^2+1}}\right )} K[1]+i \sqrt {f\left (\frac {1}{\sqrt {K[1]^2+1}}\right )-1}\right )}dK[1]=-\log (x)+c_1,y(x)\right ]\\ \text {Solve}\left [\int _1^{\frac {y(x)}{x}}\frac {f\left (\frac {1}{\sqrt {K[2]^2+1}}\right ) K[2]^2+f\left (\frac {1}{\sqrt {K[2]^2+1}}\right )-1}{\sqrt {f\left (\frac {1}{\sqrt {K[2]^2+1}}\right )} (K[2]-i) (K[2]+i) \left (\sqrt {f\left (\frac {1}{\sqrt {K[2]^2+1}}\right )} K[2]-i \sqrt {f\left (\frac {1}{\sqrt {K[2]^2+1}}\right )-1}\right )}dK[2]=-\log (x)+c_1,y(x)\right ] \end{align*}

✗ Sympy

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

Timed Out