55.2.72 problem 72

Internal problem ID [13298]
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 : 72
Date solved : Wednesday, October 01, 2025 at 06:18:26 AM
CAS classification : [_Riccati]

\begin{align*} x^{n +1} y^{\prime }&=a \,x^{2 n} y^{2}+b \,x^{n} y+c \,x^{m}+d \end{align*}

✓ Maple. Time used: 0.004 (sec). Leaf size: 285

ode:=x^(n+1)*diff(y(x),x) = a*x^(2*n)*y(x)^2+b*x^n*y(x)+c*x^m+d; 
dsolve(ode,y(x), singsol=all);
 

\[ y = -\frac {x^{-n} \left (-2 \sqrt {a c}\, \left (\operatorname {BesselY}\left (\frac {\sqrt {-4 a d +b^{2}+2 b n +n^{2}}}{m}+1, \frac {2 \sqrt {a c}\, x^{\frac {m}{2}}}{m}\right ) c_1 +\operatorname {BesselJ}\left (\frac {\sqrt {-4 a d +b^{2}+2 b n +n^{2}}}{m}+1, \frac {2 \sqrt {a c}\, x^{\frac {m}{2}}}{m}\right )\right ) x^{\frac {m}{2}}+\left (\operatorname {BesselY}\left (\frac {\sqrt {-4 a d +b^{2}+2 b n +n^{2}}}{m}, \frac {2 \sqrt {a c}\, x^{\frac {m}{2}}}{m}\right ) c_1 +\operatorname {BesselJ}\left (\frac {\sqrt {-4 a d +b^{2}+2 b n +n^{2}}}{m}, \frac {2 \sqrt {a c}\, x^{\frac {m}{2}}}{m}\right )\right ) \left (\sqrt {-4 a d +b^{2}+2 b n +n^{2}}+b +n \right )\right )}{2 a \left (\operatorname {BesselY}\left (\frac {\sqrt {-4 a d +b^{2}+2 b n +n^{2}}}{m}, \frac {2 \sqrt {a c}\, x^{\frac {m}{2}}}{m}\right ) c_1 +\operatorname {BesselJ}\left (\frac {\sqrt {-4 a d +b^{2}+2 b n +n^{2}}}{m}, \frac {2 \sqrt {a c}\, x^{\frac {m}{2}}}{m}\right )\right )} \]

✓ Mathematica. Time used: 1.085 (sec). Leaf size: 2576

ode=x^(n+1)*D[y[x],x]==a*x^(2*n)*y[x]^2+b*x^n*y[x]+c*x^m+d; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

Too large to display

✗ Sympy

from sympy import * 
x = symbols("x") 
a = symbols("a") 
b = symbols("b") 
c = symbols("c") 
d = symbols("d") 
m = symbols("m") 
n = symbols("n") 
y = Function("y") 
ode = Eq(-a*x**(2*n)*y(x)**2 - b*x**n*y(x) - c*x**m - d + x**(n + 1)*Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE -a*x**(n - 1)*y(x)**2 - b*y(x)/x - c*x**(m - n - 1) - d*x**(-n - 1) + Derivative(y(x), x) cannot be solved by the factorable group method