61.4.13 problem 34

Internal problem ID [12039]
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 : 34
Date solved : Sunday, March 30, 2025 at 10:20:36 PM
CAS classification : [[_1st_order, `_with_symmetry_[F(x),G(x)]`], _Riccati]

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

✓ Maple. Time used: 0.006 (sec). Leaf size: 38

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

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

✓ Mathematica. Time used: 1.541 (sec). Leaf size: 40

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

\begin{align*} y(x)\to \frac {\lambda }{-a e^{\lambda x}+c_1 \lambda }+b x^n+c \\ y(x)\to b x^n+c \\ \end{align*}

✗ 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*(-b*x**n - c + y(x))**2*exp(lambda_*x) - b*n*x**(n - 1) + Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

Timed Out