61.4.5 problem 26

Internal problem ID [12031]
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 : 26
Date solved : Sunday, March 30, 2025 at 10:19:44 PM
CAS classification : [_Riccati]

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

✓ Maple. Time used: 0.022 (sec). Leaf size: 88

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

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

✓ Mathematica. Time used: 3.34 (sec). Leaf size: 158

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

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

✗ Sympy

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

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