55.2.32 problem 32

Internal problem ID [13258]
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 : 32
Date solved : Wednesday, October 01, 2025 at 04:43:39 AM
CAS classification : [_Riccati]

\begin{align*} y^{\prime }&=-a n \,x^{n -1} y^{2}+c \,x^{m} \left (a \,x^{n}+b \right ) y-c \,x^{m} \end{align*}

✓ Maple. Time used: 0.034 (sec). Leaf size: 199

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

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

✓ Mathematica. Time used: 5.554 (sec). Leaf size: 304

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

\begin{align*} y(x)&\to \frac {a c_1 n \left (a x^n+b\right ) \int _1^x\frac {\exp \left (c K[1]^{m+1} \left (\frac {a K[1]^n}{m+n+1}+\frac {b}{m+1}\right )\right ) K[1]^{n-1}}{\left (a K[1]^n+b\right )^2}dK[1]+a^2 n x^n+c_1 e^{c x^{m+1} \left (\frac {a x^n}{m+n+1}+\frac {b}{m+1}\right )}+a b n}{a n \left (a x^n+b\right )^2 \left (1+c_1 \int _1^x\frac {\exp \left (c K[1]^{m+1} \left (\frac {a K[1]^n}{m+n+1}+\frac {b}{m+1}\right )\right ) K[1]^{n-1}}{\left (a K[1]^n+b\right )^2}dK[1]\right )}\\ y(x)&\to \frac {\frac {e^{c x^{m+1} \left (\frac {a x^n}{m+n+1}+\frac {b}{m+1}\right )}}{a n \int _1^x\frac {\exp \left (c K[1]^{m+1} \left (\frac {a K[1]^n}{m+n+1}+\frac {b}{m+1}\right )\right ) K[1]^{n-1}}{\left (a K[1]^n+b\right )^2}dK[1]}+a x^n+b}{\left (a x^n+b\right )^2} \end{align*}

✗ 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*n*x**(n - 1)*y(x)**2 - c*x**m*(a*x**n + b)*y(x) + c*x**m + Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

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