61.4.10 problem 31
Internal
problem
ID
[12036]
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
:
31
Date
solved
:
Wednesday, March 05, 2025 at 03:54:31 PM
CAS
classification
:
[_Riccati]
\begin{align*} y^{\prime }&=-\left (k +1\right ) x^{k} y^{2}+a \,x^{k +1} {\mathrm e}^{\lambda x} y-a \,{\mathrm e}^{\lambda x} \end{align*}
✓ Maple. Time used: 0.012 (sec). Leaf size: 184
ode:=diff(y(x),x) = -(k+1)*x^k*y(x)^2+a*x^(k+1)*exp(lambda*x)*y(x)-exp(lambda*x)*a;
dsolve(ode,y(x), singsol=all);
\[
y = \frac {x^{-k -1} \left (x^{k +1} {\mathrm e}^{\int \frac {{\mathrm e}^{\lambda x} x^{k +1} a x -2 k -2}{x}d x}+\left (\int x^{k} {\mathrm e}^{a \left (\int x^{k +1} {\mathrm e}^{\lambda x}d x \right )-2 \left (\int \frac {1}{x}d x \right ) \left (k +1\right )}d x \right ) k +\int x^{k} {\mathrm e}^{a \left (\int x^{k +1} {\mathrm e}^{\lambda x}d x \right )-2 \left (\int \frac {1}{x}d x \right ) \left (k +1\right )}d x -c_{1} \right )}{\left (\int x^{k} {\mathrm e}^{a \left (\int x^{k +1} {\mathrm e}^{\lambda x}d x \right )-2 \left (\int \frac {1}{x}d x \right ) \left (k +1\right )}d x \right ) k +\int x^{k} {\mathrm e}^{a \left (\int x^{k +1} {\mathrm e}^{\lambda x}d x \right )-2 \left (\int \frac {1}{x}d x \right ) \left (k +1\right )}d x -c_{1}}
\]
✓ Mathematica. Time used: 46.167 (sec). Leaf size: 401
ode=D[y[x],x]==-(k+1)*x^k*y[x]^2+a*x^(k+1)*Exp[\[Lambda]*x]*y[x]-a*Exp[\[Lambda]*x];
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*}
y(x)\to \frac {a \lambda \exp \left (\int _1^{e^{x \lambda }}-\frac {a \left (\frac {\log (K[1])}{\lambda }\right )^{k+1}}{\lambda }dK[1]\right ) \left (1+c_1 \int _1^{e^{x \lambda }}\exp \left (-\int _1^{K[2]}-\frac {a \left (\frac {\log (K[1])}{\lambda }\right )^{k+1}}{\lambda }dK[1]\right )dK[2]\right )}{a \lambda \left (\frac {\log \left (e^{\lambda x}\right )}{\lambda }\right )^{k+1} \exp \left (\int _1^{e^{x \lambda }}-\frac {a \left (\frac {\log (K[1])}{\lambda }\right )^{k+1}}{\lambda }dK[1]\right )+a c_1 \lambda \left (\frac {\log \left (e^{\lambda x}\right )}{\lambda }\right )^{k+1} \exp \left (\int _1^{e^{x \lambda }}-\frac {a \left (\frac {\log (K[1])}{\lambda }\right )^{k+1}}{\lambda }dK[1]\right ) \int _1^{e^{x \lambda }}\exp \left (-\int _1^{K[2]}-\frac {a \left (\frac {\log (K[1])}{\lambda }\right )^{k+1}}{\lambda }dK[1]\right )dK[2]-c_1 \lambda ^2} \\
y(x)\to \frac {a \lambda \exp \left (\int _1^{e^{x \lambda }}-\frac {a \left (\frac {\log (K[1])}{\lambda }\right )^{k+1}}{\lambda }dK[1]\right ) \int _1^{e^{x \lambda }}\exp \left (-\int _1^{K[2]}-\frac {a \left (\frac {\log (K[1])}{\lambda }\right )^{k+1}}{\lambda }dK[1]\right )dK[2]}{a \lambda \left (\frac {\log \left (e^{\lambda x}\right )}{\lambda }\right )^{k+1} \exp \left (\int _1^{e^{x \lambda }}-\frac {a \left (\frac {\log (K[1])}{\lambda }\right )^{k+1}}{\lambda }dK[1]\right ) \int _1^{e^{x \lambda }}\exp \left (-\int _1^{K[2]}-\frac {a \left (\frac {\log (K[1])}{\lambda }\right )^{k+1}}{\lambda }dK[1]\right )dK[2]-\lambda ^2} \\
\end{align*}
✗ Sympy
from sympy import *
x = symbols("x")
a = symbols("a")
k = symbols("k")
cg = symbols("cg")
y = Function("y")
ode = Eq(-a*x**(k + 1)*y(x)*exp(cg*x) + a*exp(cg*x) + x**k*(k + 1)*y(x)**2 + Derivative(y(x), x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
NotImplementedError : The given ODE -a*x**(k + 1)*y(x)*exp(cg*x) + a*exp(cg*x) + k*x**k*y(x)**2 + x**k*y(x)**2 + Derivative(y(x), x) cannot be solved by the factorable group method