61.8.7 problem 16

Internal problem ID [12088]
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 : 16
Date solved : Friday, March 14, 2025 at 04:05:02 AM
CAS classification : [_Riccati]

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

✓ Maple. Time used: 0.003 (sec). Leaf size: 99

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

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

✓ Mathematica. Time used: 1.448 (sec). Leaf size: 290

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

\[ \text {Solve}\left [\int _1^x\frac {\exp \left (-\int _1^{K[2]}\left (2 a c \log ^n(K[1])-b \log ^m(K[1])\right )dK[1]\right ) \left (-b \log ^m(K[2])+a c \log ^n(K[2])-a y(x) \log ^n(K[2])\right )}{a b (m-n) (c+y(x))}dK[2]+\int _1^{y(x)}\left (\frac {\exp \left (-\int _1^x\left (2 a c \log ^n(K[1])-b \log ^m(K[1])\right )dK[1]\right )}{a b (m-n) (c+K[3])^2}-\int _1^x\left (-\frac {\exp \left (-\int _1^{K[2]}\left (2 a c \log ^n(K[1])-b \log ^m(K[1])\right )dK[1]\right ) \log ^n(K[2])}{b (m-n) (c+K[3])}-\frac {\exp \left (-\int _1^{K[2]}\left (2 a c \log ^n(K[1])-b \log ^m(K[1])\right )dK[1]\right ) \left (-b \log ^m(K[2])+a c \log ^n(K[2])-a K[3] \log ^n(K[2])\right )}{a b (m-n) (c+K[3])^2}\right )dK[2]\right )dK[3]=c_1,y(x)\right ] \]

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

Timed Out