55.3.18 problem 18

Internal problem ID [13322]
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. Equations Containing Exponential Functions
Problem number : 18
Date solved : Wednesday, October 01, 2025 at 07:15:21 AM
CAS classification : [_Riccati]

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

✗ Maple

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

\[ \text {No solution found} \]

✓ Mathematica. Time used: 7.049 (sec). Leaf size: 3021

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

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✗ Sympy

from sympy import * 
x = symbols("x") 
a = symbols("a") 
b = symbols("b") 
c = symbols("c") 
d = symbols("d") 
k = symbols("k") 
n = symbols("n") 
y = Function("y") 
ode = Eq(-a*y(x)**2*exp(k*x) - b*y(x) - c*exp(k*n*x) - d*exp(k*x*(2*n + 1)) + Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE -a*y(x)**2*exp(k*x) - b*y(x) - c*exp(k*n*x) - d*exp(k*x*(2*n + 1)) + Derivative(y(x), x) cannot be solved by the lie group method