61.8.13 problem 22

Internal problem ID [12094]
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 : 22
Date solved : Sunday, March 30, 2025 at 10:40:57 PM
CAS classification : [_Riccati]

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

✓ Maple. Time used: 0.008 (sec). Leaf size: 107

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

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

✓ Mathematica. Time used: 1.662 (sec). Leaf size: 275

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

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

✗ Sympy

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

Timed Out