60.1.198 problem 201

Internal problem ID [10212]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, linear first order
Problem number : 201
Date solved : Wednesday, March 05, 2025 at 08:43:33 AM
CAS classification : [_Riccati]

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

✓ Maple. Time used: 0.040 (sec). Leaf size: 21

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

\[ y = i \tan \left (-i \left (\int \sqrt {f}d x \right )+c_{1} \right ) \sqrt {f} \]

✓ Mathematica. Time used: 0.259 (sec). Leaf size: 39

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

\[ y(x)\to i \sqrt {f(x)} \tan \left (i \int _1^x-\sqrt {f(K[1])}dK[1]+c_1\right ) \]

✗ Sympy

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

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