55.3.8 problem 8

Internal problem ID [13312]
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 : 8
Date solved : Wednesday, October 01, 2025 at 06:46:46 AM
CAS classification : [_Riccati]

\begin{align*} y^{\prime }&=y^{2}+a \,{\mathrm e}^{8 \lambda x}+b \,{\mathrm e}^{6 \lambda x}+c \,{\mathrm e}^{4 \lambda x}-\lambda ^{2} \end{align*}

✓ Maple. Time used: 0.078 (sec). Leaf size: 550

ode:=diff(y(x),x) = y(x)^2+a*exp(8*lambda*x)+b*exp(6*lambda*x)+c*exp(4*lambda*x)-lambda^2; 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} \text {Solution too large to show}\end{align*}

✓ Mathematica. Time used: 1.737 (sec). Leaf size: 1515

ode=D[y[x],x]==y[x]^2+a*Exp[8*\[Lambda]*x]+b*Exp[6*\[Lambda]*x]+c*Exp[4*\[Lambda]*x]-\[Lambda]^2; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} \text {Solution too large to show}\end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
a = symbols("a") 
b = symbols("b") 
c = symbols("c") 
lambda_ = symbols("lambda_") 
y = Function("y") 
ode = Eq(-a*exp(8*lambda_*x) - b*exp(6*lambda_*x) - c*exp(4*lambda_*x) + lambda_**2 - y(x)**2 + Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE -a*exp(8*lambda_*x) - b*exp(6*lambda_*x) - c*exp(4*lambda_*x) + lambda_**2 - y(x)**2 + Derivative(y(x), x) cannot be solved by the lie group method