61.4.2 problem 23
Internal
problem
ID
[12028]
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
:
23
Date
solved
:
Sunday, March 30, 2025 at 10:19:23 PM
CAS
classification
:
[[_1st_order, _with_linear_symmetries], _Riccati]
\begin{align*} y^{\prime }&=a \,{\mathrm e}^{\lambda x} y^{2}+b \,{\mathrm e}^{-\lambda x} \end{align*}
✓ Maple. Time used: 0.005 (sec). Leaf size: 64
ode:=diff(y(x),x) = a*exp(lambda*x)*y(x)^2+b*exp(-lambda*x);
dsolve(ode,y(x), singsol=all);
\[
y = -\frac {\left (\lambda ^{2}-\tan \left (\frac {\sqrt {4 \lambda ^{2} a b -\lambda ^{4}}\, \left (\lambda x +c_1 \right )}{2 \lambda ^{2}}\right ) \sqrt {4 \lambda ^{2} a b -\lambda ^{4}}\right ) {\mathrm e}^{-\lambda x}}{2 a \lambda }
\]
✓ Mathematica. Time used: 0.365 (sec). Leaf size: 123
ode=D[y[x],x]==a*Exp[\[Lambda]*x]*y[x]^2+b*Exp[-\[Lambda]*x];
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*}
y(x)\to \frac {e^{\lambda (-x)} \left (-\sqrt {\lambda ^2-4 a b}+\frac {2}{\frac {1}{\sqrt {\lambda ^2-4 a b}}+c_1 e^{x \sqrt {\lambda ^2-4 a b}}}-\lambda \right )}{2 a} \\
y(x)\to \frac {e^{\lambda (-x)} \left (4 a b-\lambda \left (\sqrt {\lambda ^2-4 a b}+\lambda \right )\right )}{2 a \sqrt {\lambda ^2-4 a b}} \\
\end{align*}
✓ Sympy. Time used: 7.055 (sec). Leaf size: 264
from sympy import *
x = symbols("x")
a = symbols("a")
b = symbols("b")
lambda_ = symbols("lambda_")
y = Function("y")
ode = Eq(-a*y(x)**2*exp(lambda_*x) - b*exp(-lambda_*x) + Derivative(y(x), x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
y{\left (x \right )} = \frac {\left (4 a b \sqrt {- \frac {1}{4 a b - \lambda _{}^{2}}} e^{\frac {\sqrt {- \frac {1}{4 a b - \lambda _{}^{2}}} \left (4 C_{1} a b - \lambda _{} \left (C_{1} \lambda _{} + 4 a b x - \lambda _{}^{2} x\right )\right )}{\lambda _{}}} + 4 a b \sqrt {- \frac {1}{4 a b - \lambda _{}^{2}}} - \lambda _{}^{2} \sqrt {- \frac {1}{4 a b - \lambda _{}^{2}}} e^{\frac {\sqrt {- \frac {1}{4 a b - \lambda _{}^{2}}} \left (4 C_{1} a b - \lambda _{} \left (C_{1} \lambda _{} + 4 a b x - \lambda _{}^{2} x\right )\right )}{\lambda _{}}} - \lambda _{}^{2} \sqrt {- \frac {1}{4 a b - \lambda _{}^{2}}} - \lambda _{} e^{\frac {\sqrt {- \frac {1}{4 a b - \lambda _{}^{2}}} \left (4 C_{1} a b - \lambda _{} \left (C_{1} \lambda _{} + 4 a b x - \lambda _{}^{2} x\right )\right )}{\lambda _{}}} + \lambda _{}\right ) e^{- \lambda _{} x}}{2 a \left (e^{\frac {\sqrt {- \frac {1}{4 a b - \lambda _{}^{2}}} \left (4 C_{1} a b - \lambda _{} \left (C_{1} \lambda _{} + 4 a b x - \lambda _{}^{2} x\right )\right )}{\lambda _{}}} - 1\right )}
\]