61.4.4 problem 25

Internal problem ID [12030]
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 : 25
Date solved : Sunday, March 30, 2025 at 10:19:37 PM
CAS classification : [_Riccati]

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

✓ Maple. Time used: 0.014 (sec). Leaf size: 88

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

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

✓ Mathematica. Time used: 0.923 (sec). Leaf size: 254

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

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

✗ Sympy

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

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