61.19.2 problem 2

Internal problem ID [12192]
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 : 2
Date solved : Wednesday, March 05, 2025 at 05:33:34 PM
CAS classification : [_Riccati]

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

✓ Maple. Time used: 0.003 (sec). Leaf size: 65

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

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

✓ Mathematica. Time used: 0.459 (sec). Leaf size: 185

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

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

✓ Sympy. Time used: 2.731 (sec). Leaf size: 68

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

\[ y{\left (x \right )} = \frac {b \left (e^{2 C_{1} b e^{a x}} + e^{2 b e^{a x} \int f{\left (x \right )} e^{- a x}\, dx}\right )}{e^{2 C_{1} b e^{a x}} - e^{2 b e^{a x} \int f{\left (x \right )} e^{- a x}\, dx}} \]