55.3.3 problem 3

Internal problem ID [13307]
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 : 3
Date solved : Wednesday, October 01, 2025 at 06:38:59 AM
CAS classification : [_Riccati]

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

✓ Maple. Time used: 0.018 (sec). Leaf size: 327

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

\[ y = \frac {-\operatorname {WhittakerM}\left (-\frac {i \sqrt {\sigma }\, b -2 \lambda \sqrt {c}}{2 \lambda \sqrt {c}}, \frac {i \sqrt {a}\, \sqrt {\sigma }}{\lambda }, \frac {2 i \sqrt {\sigma }\, \sqrt {c}\, {\mathrm e}^{\lambda x}}{\lambda }\right ) \left (i \left (\sqrt {a}\, \sqrt {c}-\frac {b}{2}\right ) \sqrt {\sigma }+\frac {\lambda \sqrt {c}}{2}\right )+c_1 \lambda \operatorname {WhittakerW}\left (-\frac {i \sqrt {\sigma }\, b -2 \lambda \sqrt {c}}{2 \lambda \sqrt {c}}, \frac {i \sqrt {a}\, \sqrt {\sigma }}{\lambda }, \frac {2 i \sqrt {\sigma }\, \sqrt {c}\, {\mathrm e}^{\lambda x}}{\lambda }\right ) \sqrt {c}+\left (-i {\mathrm e}^{\lambda x} c \sqrt {\sigma }-\frac {i \sqrt {\sigma }\, b}{2}+\frac {\lambda \sqrt {c}}{2}\right ) \left (\operatorname {WhittakerW}\left (-\frac {i \sqrt {\sigma }\, b}{2 \lambda \sqrt {c}}, \frac {i \sqrt {a}\, \sqrt {\sigma }}{\lambda }, \frac {2 i \sqrt {\sigma }\, \sqrt {c}\, {\mathrm e}^{\lambda x}}{\lambda }\right ) c_1 +\operatorname {WhittakerM}\left (-\frac {i \sqrt {\sigma }\, b}{2 \lambda \sqrt {c}}, \frac {i \sqrt {a}\, \sqrt {\sigma }}{\lambda }, \frac {2 i \sqrt {\sigma }\, \sqrt {c}\, {\mathrm e}^{\lambda x}}{\lambda }\right )\right )}{\sqrt {c}\, \sigma \left (\operatorname {WhittakerW}\left (-\frac {i \sqrt {\sigma }\, b}{2 \lambda \sqrt {c}}, \frac {i \sqrt {a}\, \sqrt {\sigma }}{\lambda }, \frac {2 i \sqrt {\sigma }\, \sqrt {c}\, {\mathrm e}^{\lambda x}}{\lambda }\right ) c_1 +\operatorname {WhittakerM}\left (-\frac {i \sqrt {\sigma }\, b}{2 \lambda \sqrt {c}}, \frac {i \sqrt {a}\, \sqrt {\sigma }}{\lambda }, \frac {2 i \sqrt {\sigma }\, \sqrt {c}\, {\mathrm e}^{\lambda x}}{\lambda }\right )\right )} \]

✓ Mathematica. Time used: 1.269 (sec). Leaf size: 1081

ode=D[y[x],x]==sigma*y[x]^2+a+b*Exp[\[Lambda]*x]+c*Exp[2*\[Lambda]*x]; 
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_") 
sigma = symbols("sigma") 
y = Function("y") 
ode = Eq(-a - b*exp(lambda_*x) - c*exp(2*lambda_*x) - sigma*y(x)**2 + Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

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