61.8.14 problem 23

Internal problem ID [12095]
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 : 23
Date solved : Sunday, March 30, 2025 at 10:41:08 PM
CAS classification : [_Riccati]

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

✓ Maple. Time used: 0.008 (sec). Leaf size: 124

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

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

✓ Mathematica. Time used: 1.808 (sec). Leaf size: 286

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

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

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

Timed Out