61.4.14 problem 35

Internal problem ID [12040]
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 : 35
Date solved : Sunday, March 30, 2025 at 10:20:42 PM
CAS classification : [_Riccati]

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

✓ Maple. Time used: 0.068 (sec). Leaf size: 38

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

\[ y = -\tan \left (a b \,x^{k} \left (\Gamma \left (k , -\lambda x \right )-\Gamma \left (k \right )\right ) \left (-\lambda x \right )^{-k}+c_1 \right ) b \,x^{k} \]

✓ Mathematica. Time used: 0.939 (sec). Leaf size: 47

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

\[ y(x)\to \sqrt {b^2} x^k \tan \left (-a \sqrt {b^2} x^k (\lambda (-x))^{-k} \Gamma (k,-x \lambda )+c_1\right ) \]

✗ Sympy

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

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