55.7.3 problem 3
Internal
problem
ID
[13374]
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, October 01, 2025 at 09:00:19 AM
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: 0.003 (sec). Leaf size: 464
ode:=x*diff(y(x),x) = a*y(x)^2+b*ln(x)^k+c*ln(x)^(2+2*k);
dsolve(ode,y(x), singsol=all);
\[
y = \frac {-\ln \left (x \right )^{k +1} \left (i \left (k +1\right ) \sqrt {c}\, \sqrt {a}-a b \right ) \left (k +3\right ) \operatorname {hypergeom}\left (\left [\frac {\left (3 k +5\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 (-\left (i \sqrt {c}\, \left (k +3\right ) \sqrt {a}-a b \right ) c_1 \ln \left (x \right )^{k +2} \operatorname {hypergeom}\left (\left [\frac {\left (3 k +7\right ) \sqrt {c}+i \sqrt {a}\, b}{\sqrt {c}\, \left (2 k +4\right )}\right ], \left [\frac {2 k +5}{k +2}\right ], \frac {2 i \sqrt {a}\, \sqrt {c}\, \ln \left (x \right )^{k +2}}{k +2}\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 ) \left (k +3\right )\right ) \left (k +1\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: 1.183 (sec). Leaf size: 806
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*} y(x)&\to -\frac {\log ^{k+1}(x) \left (\sqrt {c} c_1 (k+2) \sqrt {-(k+2)^2} \operatorname {HypergeometricU}\left (\frac {1}{2} \left (\frac {\sqrt {a} b}{\sqrt {c} \sqrt {-(k+2)^2}}+\frac {k+1}{k+2}\right ),\frac {k+1}{k+2},\frac {2 \sqrt {a} \sqrt {c} \log ^{k+2}(x)}{\sqrt {-(k+2)^2}}\right )+c_1 \left (\sqrt {a} b (k+2)+\sqrt {c} \sqrt {-(k+2)^2} (k+1)\right ) \operatorname {HypergeometricU}\left (\frac {1}{2} \left (\frac {\sqrt {a} b}{\sqrt {c} \sqrt {-(k+2)^2}}+\frac {3 k+5}{k+2}\right ),\frac {2 k+3}{k+2},\frac {2 \sqrt {a} \sqrt {c} \log ^{k+2}(x)}{\sqrt {-(k+2)^2}}\right )+\sqrt {c} (k+2) \sqrt {-(k+2)^2} \left (L_{-\frac {\sqrt {a} b}{2 \sqrt {c} \sqrt {-(k+2)^2}}-\frac {k+1}{2 k+4}}^{-\frac {1}{k+2}}\left (\frac {2 \sqrt {a} \sqrt {c} \log ^{k+2}(x)}{\sqrt {-(k+2)^2}}\right )+2 L_{-\frac {\sqrt {a} b}{2 \sqrt {c} \sqrt {-(k+2)^2}}-\frac {3 k+5}{2 k+4}}^{\frac {k+1}{k+2}}\left (\frac {2 \sqrt {a} \sqrt {c} \log ^{k+2}(x)}{\sqrt {-(k+2)^2}}\right )\right )\right )}{\sqrt {a} (k+2)^2 \left (L_{-\frac {\sqrt {a} b}{2 \sqrt {c} \sqrt {-(k+2)^2}}-\frac {k+1}{2 k+4}}^{-\frac {1}{k+2}}\left (\frac {2 \sqrt {a} \sqrt {c} \log ^{k+2}(x)}{\sqrt {-(k+2)^2}}\right )+c_1 \operatorname {HypergeometricU}\left (\frac {1}{2} \left (\frac {\sqrt {a} b}{\sqrt {c} \sqrt {-(k+2)^2}}+\frac {k+1}{k+2}\right ),\frac {k+1}{k+2},\frac {2 \sqrt {a} \sqrt {c} \log ^{k+2}(x)}{\sqrt {-(k+2)^2}}\right )\right )}\\ y(x)&\to \frac {\log ^{k+1}(x) \left (-\frac {\left (\sqrt {a} b (k+2)+\sqrt {c} \sqrt {-(k+2)^2} (k+1)\right ) \operatorname {HypergeometricU}\left (\frac {1}{2} \left (\frac {\sqrt {a} b}{\sqrt {c} \sqrt {-(k+2)^2}}+\frac {k+1}{k+2}+2\right ),\frac {2 k+3}{k+2},\frac {2 \sqrt {a} \sqrt {c} \log ^{k+2}(x)}{\sqrt {-(k+2)^2}}\right )}{\operatorname {HypergeometricU}\left (\frac {1}{2} \left (\frac {\sqrt {a} b}{\sqrt {c} \sqrt {-(k+2)^2}}+\frac {k+1}{k+2}\right ),\frac {k+1}{k+2},\frac {2 \sqrt {a} \sqrt {c} \log ^{k+2}(x)}{\sqrt {-(k+2)^2}}\right )}-\sqrt {c} \sqrt {-(k+2)^2} (k+2)\right )}{\sqrt {a} (k+2)^2} \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