61.3.3 problem 3
Internal
problem
ID
[12008]
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
:
Sunday, March 30, 2025 at 10:15:12 PM
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: 1.854 (sec). Leaf size: 842
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