61.3.11 problem 11
Internal
problem
ID
[12016]
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
:
11
Date
solved
:
Sunday, March 30, 2025 at 10:16:41 PM
CAS
classification
:
[[_1st_order, _with_linear_symmetries], _Riccati]
\begin{align*} y^{\prime }&=a \,{\mathrm e}^{\lambda x} y^{2}+b y+c \,{\mathrm e}^{-\lambda x} \end{align*}
✓ Maple. Time used: 0.005 (sec). Leaf size: 96
ode:=diff(y(x),x) = a*exp(lambda*x)*y(x)^2+b*y(x)+c*exp(-lambda*x);
dsolve(ode,y(x), singsol=all);
\[
y = -\frac {\left (-\sqrt {\left (b +\lambda \right )^{2} \left (4 a c -b^{2}-2 b \lambda -\lambda ^{2}\right )}\, \tan \left (\frac {\left (\left (b +\lambda \right ) x +c_1 \right ) \sqrt {\left (b +\lambda \right )^{2} \left (4 a c -b^{2}-2 b \lambda -\lambda ^{2}\right )}}{2 \left (b +\lambda \right )^{2}}\right )+\left (b +\lambda \right )^{2}\right ) {\mathrm e}^{-\lambda x}}{2 a \left (b +\lambda \right )}
\]
✓ Mathematica. Time used: 0.581 (sec). Leaf size: 188
ode=D[y[x],x]==a*Exp[\[Lambda]*x]*y[x]^2+b*y[x]+c*Exp[-\[Lambda]*x];
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*}
y(x)\to \frac {e^{\lambda (-x)} \left (-\sqrt {-4 a c+b^2+2 b \lambda +\lambda ^2}+\frac {2}{\frac {1}{\sqrt {-4 a c+b^2+2 b \lambda +\lambda ^2}}+c_1 e^{x \sqrt {-4 a c+b^2+2 b \lambda +\lambda ^2}}}-b-\lambda \right )}{2 a} \\
y(x)\to -\frac {e^{\lambda (-x)} \left (b \left (\sqrt {-4 a c+b^2+2 b \lambda +\lambda ^2}+2 \lambda \right )+\lambda \left (\sqrt {-4 a c+b^2+2 b \lambda +\lambda ^2}+\lambda \right )-4 a c+b^2\right )}{2 a \sqrt {-4 a c+b^2+2 b \lambda +\lambda ^2}} \\
\end{align*}
✓ Sympy. Time used: 85.257 (sec). Leaf size: 372
from sympy import *
x = symbols("x")
a = symbols("a")
b = symbols("b")
c = symbols("c")
lambda_ = symbols("lambda_")
y = Function("y")
ode = Eq(-a*y(x)**2*exp(lambda_*x) - b*y(x) - c*exp(-lambda_*x) + Derivative(y(x), x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
y{\left (x \right )} = \frac {\left (4 a c \sqrt {\frac {1}{- 4 a c + b^{2} + 2 b \lambda _{} + \lambda _{}^{2}}} - b^{2} \sqrt {\frac {1}{- 4 a c + b^{2} + 2 b \lambda _{} + \lambda _{}^{2}}} - 2 b \lambda _{} \sqrt {\frac {1}{- 4 a c + b^{2} + 2 b \lambda _{} + \lambda _{}^{2}}} - b - \lambda _{}^{2} \sqrt {\frac {1}{- 4 a c + b^{2} + 2 b \lambda _{} + \lambda _{}^{2}}} - \lambda _{} + \left (4 a c \sqrt {\frac {1}{- 4 a c + b^{2} + 2 b \lambda _{} + \lambda _{}^{2}}} - b^{2} \sqrt {\frac {1}{- 4 a c + b^{2} + 2 b \lambda _{} + \lambda _{}^{2}}} - 2 b \lambda _{} \sqrt {\frac {1}{- 4 a c + b^{2} + 2 b \lambda _{} + \lambda _{}^{2}}} + b - \lambda _{}^{2} \sqrt {\frac {1}{- 4 a c + b^{2} + 2 b \lambda _{} + \lambda _{}^{2}}} + \lambda _{}\right ) e^{\frac {\left (- 4 C_{1} a c + C_{1} b^{2} + \lambda _{} \left (2 C_{1} b + C_{1} \lambda _{} + 4 a c x - b^{2} x - 2 b \lambda _{} x - \lambda _{}^{2} x\right )\right ) \sqrt {\frac {1}{- 4 a c + b^{2} + 2 b \lambda _{} + \lambda _{}^{2}}}}{\lambda _{}}}\right ) e^{- \lambda _{} x}}{2 a \left (1 - e^{\frac {\left (- 4 C_{1} a c + C_{1} b^{2} + \lambda _{} \left (2 C_{1} b + C_{1} \lambda _{} + 4 a c x - b^{2} x - 2 b \lambda _{} x - \lambda _{}^{2} x\right )\right ) \sqrt {\frac {1}{- 4 a c + b^{2} + 2 b \lambda _{} + \lambda _{}^{2}}}}{\lambda _{}}}\right )}
\]