2.8.5 Problem 14

Internal problem ID [13357]
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 : 14
Date solved : Sunday, January 18, 2026 at 07:29:22 PM
CAS classification : [_linear]

2.8.5.1 Solved using first_order_ode_linear

0.154 (sec)

Entering first order ode linear solver

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

In canonical form a linear first order is

\begin{align*} y^{\prime } + q(x)y &= p(x) \end{align*}

Comparing the above to the given ode shows that

\begin{align*} q(x) &=-\ln \left (x \right )^{n} a \left (1-\ln \left (x \right ) b x \right )\\ p(x) &=b \left (\ln \left (x \right )+1\right ) \end{align*}

The integrating factor \(\mu \) is

\begin{align*} \mu &= e^{\int {q\,dx}}\\ &= {\mathrm e}^{\int -\ln \left (x \right )^{n} a \left (1-\ln \left (x \right ) b x \right )d x}\\ &= {\mathrm e}^{\int -\ln \left (x \right )^{n} a \left (1-\ln \left (x \right ) b x \right )d x} \end{align*}

The ode becomes

\begin{align*} \frac {\mathop {\mathrm {d}}}{ \mathop {\mathrm {d}x}}\left ( \mu y\right ) &= \mu p \\ \frac {\mathop {\mathrm {d}}}{ \mathop {\mathrm {d}x}}\left ( \mu y\right ) &= \left (\mu \right ) \left (b \left (\ln \left (x \right )+1\right )\right ) \\ \frac {\mathop {\mathrm {d}}}{ \mathop {\mathrm {d}x}} \left (y \,{\mathrm e}^{\int -\ln \left (x \right )^{n} a \left (1-\ln \left (x \right ) b x \right )d x}\right ) &= \left ({\mathrm e}^{\int -\ln \left (x \right )^{n} a \left (1-\ln \left (x \right ) b x \right )d x}\right ) \left (b \left (\ln \left (x \right )+1\right )\right ) \\ \mathrm {d} \left (y \,{\mathrm e}^{\int -\ln \left (x \right )^{n} a \left (1-\ln \left (x \right ) b x \right )d x}\right ) &= \left (b \left (\ln \left (x \right )+1\right ) {\mathrm e}^{\int -\ln \left (x \right )^{n} a \left (1-\ln \left (x \right ) b x \right )d x}\right )\, \mathrm {d} x \\ \end{align*}

Integrating gives

\begin{align*} y \,{\mathrm e}^{\int -\ln \left (x \right )^{n} a \left (1-\ln \left (x \right ) b x \right )d x}&= \int {b \left (\ln \left (x \right )+1\right ) {\mathrm e}^{\int -\ln \left (x \right )^{n} a \left (1-\ln \left (x \right ) b x \right )d x} \,dx} \\ &=\int b \left (\ln \left (x \right )+1\right ) {\mathrm e}^{\int -\ln \left (x \right )^{n} a \left (1-\ln \left (x \right ) b x \right )d x}d x + c_1 \end{align*}

Dividing throughout by the integrating factor \({\mathrm e}^{\int -\ln \left (x \right )^{n} a \left (1-\ln \left (x \right ) b x \right )d x}\) gives the final solution

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

Summary of solutions found

\begin{align*} y &= {\mathrm e}^{\int \ln \left (x \right )^{n} a \left (1-\ln \left (x \right ) b x \right )d x} \left (\int b \left (\ln \left (x \right )+1\right ) {\mathrm e}^{\int -\ln \left (x \right )^{n} a \left (1-\ln \left (x \right ) b x \right )d x}d x +c_1 \right ) \\ \end{align*}

2.8.5.2 ✓ Maple. Time used: 0.002 (sec). Leaf size: 51

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

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

Maple trace

Methods for first order ODEs: 
--- Trying classification methods --- 
trying a quadrature 
trying 1st order linear 
<- 1st order linear successful
 

2.8.5.3 ✓ Mathematica. Time used: 0.134 (sec). Leaf size: 78

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

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

2.8.5.4 ✗ Sympy

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

Timed Out
 

Python version: 3.12.3 (main, Aug 14 2025, 17:47:21) [GCC 13.3.0] 
Sympy version 1.14.0