61.7.3 problem 3
Internal
problem
ID
[12075]
Book
:
Handbook
of
exact
solutions
for
ordinary
differential
equations.
By
Polyanin
and
Zaitsev.
Second
edition
Section
:
Chapter
1,
section
1.2.
Riccati
Equation.
subsection
1.2.5-1.
Equations
Containing
Logarithmic
Functions
Problem
number
:
3
Date
solved
:
Wednesday, March 05, 2025 at 04:13:29 PM
CAS
classification
:
[_Riccati]
\begin{align*} x y^{\prime }&=a y^{2}+b \ln \left (x \right )^{k}+c \ln \left (x \right )^{2 k +2} \end{align*}
✓ Maple. Time used: 0.005 (sec). Leaf size: 464
ode:=x*diff(y(x),x) = y(x)^2*a+b*ln(x)^k+c*ln(x)^(2*k+2);
dsolve(ode,y(x), singsol=all);
\[
y = \frac {-\left (i \left (k +1\right ) \sqrt {c}\, \sqrt {a}-a b \right ) \left (k +3\right ) \ln \left (x \right )^{k +1} \operatorname {hypergeom}\left (\left [\frac {\left (5+3 k \right ) \sqrt {c}+i \sqrt {a}\, b}{\sqrt {c}\, \left (2 k +4\right )}\right ], \left [\frac {3+2 k}{k +2}\right ], \frac {2 i \sqrt {a}\, \sqrt {c}\, \ln \left (x \right )^{k +2}}{k +2}\right )+\left (k +1\right ) \left (-\ln \left (x \right )^{k +2} \left (i \left (k +3\right ) \sqrt {c}\, \sqrt {a}-a b \right ) c_{1} \operatorname {hypergeom}\left (\left [\frac {\left (7+3 k \right ) \sqrt {c}+i \sqrt {a}\, b}{\sqrt {c}\, \left (2 k +4\right )}\right ], \left [\frac {5+2 k}{k +2}\right ], \frac {2 i \sqrt {a}\, \sqrt {c}\, \ln \left (x \right )^{k +2}}{k +2}\right )+\left (k +3\right ) \left (\left (i \sqrt {a}\, \sqrt {c}\, \ln \left (x \right )^{k +2}-1\right ) c_{1} \operatorname {hypergeom}\left (\left [\frac {\left (k +3\right ) \sqrt {c}+i \sqrt {a}\, b}{\sqrt {c}\, \left (2 k +4\right )}\right ], \left [\frac {k +3}{k +2}\right ], \frac {2 i \sqrt {a}\, \sqrt {c}\, \ln \left (x \right )^{k +2}}{k +2}\right )+i \ln \left (x \right )^{k +1} \sqrt {a}\, \sqrt {c}\, \operatorname {hypergeom}\left (\left [\frac {\left (k +1\right ) \sqrt {c}+i \sqrt {a}\, b}{\sqrt {c}\, \left (2 k +4\right )}\right ], \left [\frac {k +1}{k +2}\right ], \frac {2 i \sqrt {a}\, \sqrt {c}\, \ln \left (x \right )^{k +2}}{k +2}\right )\right )\right )}{\left (\ln \left (x \right ) \operatorname {hypergeom}\left (\left [\frac {\left (k +3\right ) \sqrt {c}+i \sqrt {a}\, b}{\sqrt {c}\, \left (2 k +4\right )}\right ], \left [\frac {k +3}{k +2}\right ], \frac {2 i \sqrt {a}\, \sqrt {c}\, \ln \left (x \right )^{k +2}}{k +2}\right ) c_{1} +\operatorname {hypergeom}\left (\left [\frac {\left (k +1\right ) \sqrt {c}+i \sqrt {a}\, b}{\sqrt {c}\, \left (2 k +4\right )}\right ], \left [\frac {k +1}{k +2}\right ], \frac {2 i \sqrt {a}\, \sqrt {c}\, \ln \left (x \right )^{k +2}}{k +2}\right )\right ) a \left (k +3\right ) \left (k +1\right )}
\]
✓ Mathematica. Time used: 2.006 (sec). Leaf size: 1040
ode=x*D[y[x],x]==a*y[x]^2+b*(Log[x])^k+c*(Log[x])^(2*k+2);
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")
k = symbols("k")
y = Function("y")
ode = Eq(-a*y(x)**2 - b*log(x)**k - c*log(x)**(2*k + 2) + x*Derivative(y(x), x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
NotImplementedError : The given ODE Derivative(y(x), x) - (a*y(x)**2 + b*log(x)**k + c*log(x)**(2*k + 2))/x cannot be solved by the factorable group method