61.2.34 problem 34

Internal problem ID [11961]
Book : Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev. Second edition
Section : Chapter 1, section 1.2. Riccati Equation. 1.2.2. Equations Containing Power Functions
Problem number : 34
Date solved : Sunday, March 30, 2025 at 09:29:17 PM
CAS classification : [_rational, _Riccati]

\begin{align*} x y^{\prime }&=a y^{2}+b y+c \,x^{2 b} \end{align*}

✓ Maple. Time used: 0.008 (sec). Leaf size: 34

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

\[ y = \frac {\tan \left (\frac {x^{b} \sqrt {a}\, \sqrt {c}-c_1 b}{b}\right ) \sqrt {c}\, x^{b}}{\sqrt {a}} \]

✓ Mathematica. Time used: 0.344 (sec). Leaf size: 139

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

\begin{align*} y(x)\to \frac {\sqrt {c} x^b \left (-\cos \left (\frac {\sqrt {a} \sqrt {c} x^b}{b}\right )+c_1 \sin \left (\frac {\sqrt {a} \sqrt {c} x^b}{b}\right )\right )}{\sqrt {a} \left (\sin \left (\frac {\sqrt {a} \sqrt {c} x^b}{b}\right )+c_1 \cos \left (\frac {\sqrt {a} \sqrt {c} x^b}{b}\right )\right )} \\ y(x)\to \frac {\sqrt {c} x^b \tan \left (\frac {\sqrt {a} \sqrt {c} x^b}{b}\right )}{\sqrt {a}} \\ \end{align*}

✗ Sympy

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

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