61.20.1 problem 34

Internal problem ID [12224]
Book : Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev. Second edition
Section : Chapter 1, section 1.2. Riccati Equation. subsection 1.2.8-2. Equations containing arbitrary functions and their derivatives.
Problem number : 34
Date solved : Wednesday, March 05, 2025 at 06:09:09 PM
CAS classification : [_Riccati]

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

✓ Maple. Time used: 0.003 (sec). Leaf size: 45

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

\[ y = \frac {-f \left (\int {\mathrm e}^{2 \left (\int fd x \right )}d x \right )+f c_{1} +{\mathrm e}^{2 \left (\int fd x \right )}}{c_{1} -\int {\mathrm e}^{2 \left (\int fd x \right )}d x} \]

✗ Mathematica

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

Not solved

✗ Sympy

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

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