60.2.279 problem 856

Internal problem ID [10853]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 856
Date solved : Sunday, March 30, 2025 at 07:15:15 PM
CAS classification : [NONE]

\begin{align*} y^{\prime }&=-\frac {\left (-\frac {1}{x}-\textit {\_F1} \left (y^{2}-2 x \right )\right ) x}{\sqrt {y^{2}}} \end{align*}

✓ Maple. Time used: 0.076 (sec). Leaf size: 65

ode:=diff(y(x),x) = -(-1/x-_F1(y(x)^2-2*x))/(y(x)^2)^(1/2)*x; 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= \sqrt {2 \operatorname {RootOf}\left (x^{2}-2 \int _{}^{\textit {\_Z}}\frac {1}{\textit {\_F1} \left (2 \textit {\_a} \right )}d \textit {\_a} +4 c_1 \right )+2 x} \\ y &= -\sqrt {2 \operatorname {RootOf}\left (x^{2}-2 \int _{}^{\textit {\_Z}}\frac {1}{\textit {\_F1} \left (2 \textit {\_a} \right )}d \textit {\_a} +4 c_1 \right )+2 x} \\ \end{align*}

✓ Mathematica. Time used: 0.308 (sec). Leaf size: 99

ode=D[y[x],x] == (x*(x^(-1) + F1[-2*x + y[x]^2]))/Sqrt[y[x]^2]; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ \text {Solve}\left [\int _1^{y(x)}\left (\frac {\sqrt {K[2]^2}}{\text {F1}\left (K[2]^2-2 x\right )}-\int _1^x\frac {2 K[2] \text {F1}''\left (K[2]^2-2 K[1]\right )}{\text {F1}\left (K[2]^2-2 K[1]\right )^2}dK[1]\right )dK[2]+\int _1^x\left (-K[1]-\frac {1}{\text {F1}\left (y(x)^2-2 K[1]\right )}\right )dK[1]=c_1,y(x)\right ] \]

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
F1 = Function("F1") 
ode = Eq(x*(-F1(-2*x + y(x)**2) - 1/x)/sqrt(y(x)**2) + Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE -(x*F1(-2*x + y(x)**2) + 1)/sqrt(y(x)**2) + Derivative(y(x), x) cannot be solved by the factorable group method