61.8.4 problem 13

Internal problem ID [12085]
Book : Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev. Second edition
Section : Chapter 1, section 1.2. Riccati Equation. subsection 1.2.5-2
Problem number : 13
Date solved : Wednesday, March 05, 2025 at 04:14:47 PM
CAS classification : [_Riccati]

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

✓ Maple. Time used: 0.014 (sec). Leaf size: 184

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

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

✓ Mathematica. Time used: 2.689 (sec). Leaf size: 248

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

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

✗ Sympy

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

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