55.2.30 problem 30

Internal problem ID [13256]
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 : 30
Date solved : Wednesday, October 01, 2025 at 04:37:37 AM
CAS classification : [_Riccati]

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

✓ Maple. Time used: 0.006 (sec). Leaf size: 88

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

\[ c_1 +a \int _{}^{x}\textit {\_a}^{n} {\mathrm e}^{-\frac {2 \left (-\frac {b \,\textit {\_a}^{m} \left (n +1\right )}{2}+a c \,\textit {\_a}^{n} \left (m +1\right )\right ) \textit {\_a}}{\left (m +1\right ) \left (n +1\right )}}d \textit {\_a} +\frac {{\mathrm e}^{-\frac {2 \left (-\frac {x^{m} b \left (n +1\right )}{2}+x^{n} a c \left (m +1\right )\right ) x}{\left (m +1\right ) \left (n +1\right )}}}{c +y} = 0 \]

✓ Mathematica. Time used: 0.951 (sec). Leaf size: 286

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

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

✗ Sympy

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

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