61.19.9 problem 9

Internal problem ID [12199]
Book : Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev. Second edition
Section : Chapter 1, section 1.2. Riccati Equation. subsection 1.2.8-1. Equations containing arbitrary functions (but not containing their derivatives).
Problem number : 9
Date solved : Wednesday, March 05, 2025 at 05:34:07 PM
CAS classification : [_Riccati]

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

✓ Maple. Time used: 0.002 (sec). Leaf size: 67

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

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

✓ Mathematica. Time used: 0.514 (sec). Leaf size: 201

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

\[ \text {Solve}\left [\int _1^x-\frac {\exp \left (-\int _1^{K[2]}(-2 a f(K[1])-g(K[1]))dK[1]\right ) (a f(K[2])+y(x) f(K[2])+g(K[2]))}{a-y(x)}dK[2]+\int _1^{y(x)}\left (-\int _1^x\left (-\frac {\exp \left (-\int _1^{K[2]}(-2 a f(K[1])-g(K[1]))dK[1]\right ) f(K[2])}{a-K[3]}-\frac {\exp \left (-\int _1^{K[2]}(-2 a f(K[1])-g(K[1]))dK[1]\right ) (a f(K[2])+K[3] f(K[2])+g(K[2]))}{(a-K[3])^2}\right )dK[2]-\frac {\exp \left (-\int _1^x(-2 a f(K[1])-g(K[1]))dK[1]\right )}{(K[3]-a)^2}\right )dK[3]=c_1,y(x)\right ] \]

✗ Sympy

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

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