61.4.12 problem 33

Internal problem ID [12038]
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 : 33
Date solved : Sunday, March 30, 2025 at 10:20:29 PM
CAS classification : [_Riccati]

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

✓ Maple. Time used: 0.007 (sec). Leaf size: 104

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

\[ y = \frac {\left (\tan \left (\frac {\left (b \,x^{n} \left (\Gamma \left (n , -\lambda x \right ) n -\Gamma \left (n +1\right )\right ) \left (-\lambda x \right )^{-n}+b \,x^{n} {\mathrm e}^{\lambda x}+c_1 \lambda \right ) \sqrt {4 b^{2} a c -b^{4}}}{2 b^{2} \lambda }\right ) \sqrt {4 b^{2} a c -b^{4}}-b^{2}\right ) {\mathrm e}^{-\lambda x}}{2 a b} \]

✓ Mathematica. Time used: 1.486 (sec). Leaf size: 102

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

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

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

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