61.7.3 problem 3
Internal
problem
ID
[12154]
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
:
Tuesday, January 28, 2025 at 12:44:57 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*}
✓ Solution by Maple
Time used: 0.005 (sec). Leaf size: 464
dsolve(x*diff(y(x),x)=a*y(x)^2+b*(ln(x))^k+c*(ln(x))^(2*k+2),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 )}
\]
✓ Solution by Mathematica
Time used: 2.006 (sec). Leaf size: 1040
DSolve[x*D[y[x],x]==a*y[x]^2+b*(Log[x])^k+c*(Log[x])^(2*k+2),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 {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 )}{\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} \\
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*}