61.20.7 problem 40
Internal
problem
ID
[12230]
Book
:
Handbook
of
exact
solutions
for
ordinary
differential
equations.
By
Polyanin
and
Zaitsev.
Second
edition
Section
:
Chapter
1,
section
1.2.
Riccati
Equation.
subsection
1.2.8-2.
Equations
containing
arbitrary
functions
and
their
derivatives.
Problem
number
:
40
Date
solved
:
Monday, March 31, 2025 at 04:39:58 AM
CAS
classification
:
[_Riccati]
\begin{align*} y^{\prime }&=f^{\prime }\left (x \right ) y^{2}+a \,{\mathrm e}^{\lambda x} f \left (x \right ) y+a \,{\mathrm e}^{\lambda x} \end{align*}
✓ Maple. Time used: 0.025 (sec). Leaf size: 114
ode:=diff(y(x),x) = diff(f(x),x)*y(x)^2+a*exp(lambda*x)*f(x)*y(x)+exp(lambda*x)*a;
dsolve(ode,y(x), singsol=all);
\[
y = \frac {-f \left (x \right ) {\mathrm e}^{\int \frac {{\mathrm e}^{\lambda x} f \left (x \right )^{2} a -2 f^{\prime }\left (x \right )}{f \left (x \right )}d x}-\int f^{\prime }\left (x \right ) {\mathrm e}^{a \int {\mathrm e}^{\lambda x} f \left (x \right )d x -2 \int \frac {f^{\prime }\left (x \right )}{f \left (x \right )}d x}d x -c_1}{f \left (x \right ) \left (c_1 +\int f^{\prime }\left (x \right ) {\mathrm e}^{a \int {\mathrm e}^{\lambda x} f \left (x \right )d x -2 \int \frac {f^{\prime }\left (x \right )}{f \left (x \right )}d x}d x \right )}
\]
✓ Mathematica. Time used: 43.857 (sec). Leaf size: 319
ode=D[y[x],x]==D[ f[x],x]*y[x]^2+a*Exp[\[Lambda]*x]*f[x]*y[x]+a*Exp[\[Lambda]*x];
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*}
y(x)\to -\frac {a \exp \left (\int _1^{e^{x \lambda }}-\frac {a f\left (\frac {\log (K[1])}{\lambda }\right )}{\lambda }dK[1]\right ) \left (1+c_1 \int _1^{e^{x \lambda }}\exp \left (-\int _1^{K[2]}-\frac {a f\left (\frac {\log (K[1])}{\lambda }\right )}{\lambda }dK[1]\right )dK[2]\right )}{a f\left (\frac {\log \left (e^{\lambda x}\right )}{\lambda }\right ) \exp \left (\int _1^{e^{x \lambda }}-\frac {a f\left (\frac {\log (K[1])}{\lambda }\right )}{\lambda }dK[1]\right ) \left (1+c_1 \int _1^{e^{x \lambda }}\exp \left (-\int _1^{K[2]}-\frac {a f\left (\frac {\log (K[1])}{\lambda }\right )}{\lambda }dK[1]\right )dK[2]\right )-c_1 \lambda } \\
y(x)\to \frac {a \exp \left (\int _1^{e^{x \lambda }}-\frac {a f\left (\frac {\log (K[1])}{\lambda }\right )}{\lambda }dK[1]\right ) \int _1^{e^{x \lambda }}\exp \left (-\int _1^{K[2]}-\frac {a f\left (\frac {\log (K[1])}{\lambda }\right )}{\lambda }dK[1]\right )dK[2]}{\lambda -a f\left (\frac {\log \left (e^{\lambda x}\right )}{\lambda }\right ) \exp \left (\int _1^{e^{x \lambda }}-\frac {a f\left (\frac {\log (K[1])}{\lambda }\right )}{\lambda }dK[1]\right ) \int _1^{e^{x \lambda }}\exp \left (-\int _1^{K[2]}-\frac {a f\left (\frac {\log (K[1])}{\lambda }\right )}{\lambda }dK[1]\right )dK[2]} \\
\end{align*}
✗ Sympy
from sympy import *
x = symbols("x")
a = symbols("a")
lambda_ = symbols("lambda_")
y = Function("y")
f = Function("f")
ode = Eq(-a*f(x)*y(x)*exp(lambda_*x) - a*exp(lambda_*x) - y(x)**2*Derivative(f(x), x) + Derivative(y(x), x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
NotImplementedError : The given ODE -a*f(x)*y(x)*exp(lambda_*x) - a*exp(lambda_*x) - y(x)**2*Derivative(f(x), x) + Derivative(y(x), x) cannot be solved by the lie group method