61.4.6 problem 27

Internal problem ID [12032]
Book : Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev. Second edition
Section : Chapter 1, section 1.2. Riccati Equation. subsection 1.2.3-2. Equations with power and exponential functions
Problem number : 27
Date solved : Sunday, March 30, 2025 at 10:19:48 PM
CAS classification : [_Riccati]

\begin{align*} y^{\prime }&=a \,{\mathrm e}^{\lambda x} y^{2}-a b \,x^{n} {\mathrm e}^{\lambda x} y+b n \,x^{n -1} \end{align*}

✓ Maple. Time used: 0.023 (sec). Leaf size: 58

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

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

✓ Mathematica. Time used: 3.332 (sec). Leaf size: 260

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

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

✗ Sympy

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

NotImplementedError : The given ODE a*b*x**n*y(x)*exp(lambda_*x) - a*y(x)**2*exp(lambda_*x) - b*n*x**(n - 1) + Derivative(y(x), x) cannot be solved by the lie group method