61.7.7 problem 7
Internal
problem
ID
[12158]
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
:
7
Date
solved
:
Tuesday, January 28, 2025 at 12:45:40 AM
CAS
classification
:
[_Riccati]
\begin{align*} x^{2} y^{\prime }&=x^{2} y^{2}+a \ln \left (x \right )^{2}+b \ln \left (x \right )+c \end{align*}
✓ Solution by Maple
Time used: 0.003 (sec). Leaf size: 454
dsolve(x^2*diff(y(x),x)=x^2*y(x)^2+a*(ln(x))^2+b*ln(x)+c,y(x), singsol=all)
\[
y = \frac {\left (-\frac {\ln \left (x \right ) b \left (\ln \left (x \right ) b -4 c +1\right ) a^{{5}/{2}}}{12}-\frac {\left (\ln \left (x \right ) b -c +\frac {1}{4}\right ) b^{2} a^{{3}/{2}}}{12}+\frac {\left (c -\frac {1}{4}\right ) \ln \left (x \right )^{2} a^{{7}/{2}}}{3}-\frac {\sqrt {a}\, b^{4}}{48}-i a^{2} \left (a \ln \left (x \right )+\frac {b}{2}\right )^{2}\right ) c_{1} \operatorname {hypergeom}\left (\left [\frac {28 a^{{3}/{2}}+i \left (4 c -1\right ) a -i b^{2}}{16 a^{{3}/{2}}}\right ], \left [\frac {5}{2}\right ], \frac {i \left (2 a \ln \left (x \right )+b \right )^{2}}{4 a^{{3}/{2}}}\right )+\left (-\frac {a^{{5}/{2}} b}{4}+\left (-\frac {\ln \left (x \right )}{2}-1\right ) a^{{7}/{2}}+i a^{2} \left (a \ln \left (x \right )+\frac {b}{2}\right )^{2}\right ) c_{1} \operatorname {hypergeom}\left (\left [\frac {12 a^{{3}/{2}}+i \left (4 c -1\right ) a -i b^{2}}{16 a^{{3}/{2}}}\right ], \left [\frac {3}{2}\right ], \frac {i \left (2 a \ln \left (x \right )+b \right )^{2}}{4 a^{{3}/{2}}}\right )+\frac {\left (\left (c -\frac {1}{4}\right ) \ln \left (x \right ) a^{{5}/{2}}-\frac {\left (\ln \left (x \right ) b -2 c +\frac {1}{2}\right ) b \,a^{{3}/{2}}}{4}-\frac {\sqrt {a}\, b^{3}}{8}-i a^{2} \left (a \ln \left (x \right )+\frac {b}{2}\right )\right ) \operatorname {hypergeom}\left (\left [\frac {20 a^{{3}/{2}}+i \left (4 c -1\right ) a -i b^{2}}{16 a^{{3}/{2}}}\right ], \left [\frac {3}{2}\right ], \frac {i \left (2 a \ln \left (x \right )+b \right )^{2}}{4 a^{{3}/{2}}}\right )}{2}+\frac {\operatorname {hypergeom}\left (\left [\frac {4 a^{{3}/{2}}+i \left (4 c -1\right ) a -i b^{2}}{16 a^{{3}/{2}}}\right ], \left [\frac {1}{2}\right ], \frac {i \left (2 a \ln \left (x \right )+b \right )^{2}}{4 a^{{3}/{2}}}\right ) \left (i \ln \left (x \right ) a^{3}+\frac {i a^{2} b}{2}-\frac {a^{{5}/{2}}}{2}\right )}{2}}{a^{{5}/{2}} x \left (c_{1} \left (a \ln \left (x \right )+\frac {b}{2}\right ) \operatorname {hypergeom}\left (\left [\frac {12 a^{{3}/{2}}+i \left (4 c -1\right ) a -i b^{2}}{16 a^{{3}/{2}}}\right ], \left [\frac {3}{2}\right ], \frac {i \left (2 a \ln \left (x \right )+b \right )^{2}}{4 a^{{3}/{2}}}\right )+\frac {\operatorname {hypergeom}\left (\left [\frac {4 a^{{3}/{2}}+i \left (4 c -1\right ) a -i b^{2}}{16 a^{{3}/{2}}}\right ], \left [\frac {1}{2}\right ], \frac {i \left (2 a \ln \left (x \right )+b \right )^{2}}{4 a^{{3}/{2}}}\right )}{2}\right )}
\]
✓ Solution by Mathematica
Time used: 0.615 (sec). Leaf size: 868
DSolve[x^2*D[y[x],x]==x^2*y[x]^2+a*(Log[x])^2+b*Log[x]+c,y[x],x,IncludeSingularSolutions -> True]
\begin{align*}
y(x)\to \frac {i b \operatorname {ParabolicCylinderD}\left (\frac {-i b^2-4 a^{3/2}+i a (4 c-1)}{8 a^{3/2}},-\frac {\left (\frac {1}{2}-\frac {i}{2}\right ) (b+2 a \log (x))}{a^{3/4}}\right )+2 i a \log (x) \operatorname {ParabolicCylinderD}\left (\frac {-i b^2-4 a^{3/2}+i a (4 c-1)}{8 a^{3/2}},-\frac {\left (\frac {1}{2}-\frac {i}{2}\right ) (b+2 a \log (x))}{a^{3/4}}\right )-\sqrt {a} \operatorname {ParabolicCylinderD}\left (\frac {-i b^2-4 a^{3/2}+i a (4 c-1)}{8 a^{3/2}},-\frac {\left (\frac {1}{2}-\frac {i}{2}\right ) (b+2 a \log (x))}{a^{3/4}}\right )+2 (-1)^{3/4} \sqrt {2} a^{3/4} \operatorname {ParabolicCylinderD}\left (\frac {-i b^2+4 a^{3/2}+i a (4 c-1)}{8 a^{3/2}},-\frac {\left (\frac {1}{2}-\frac {i}{2}\right ) (b+2 a \log (x))}{a^{3/4}}\right )-i c_1 \left (2 a \log (x)-i \sqrt {a}+b\right ) \operatorname {ParabolicCylinderD}\left (\frac {i b^2-4 a^{3/2}+i a-4 i a c}{8 a^{3/2}},\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) (b+2 a \log (x))}{a^{3/4}}\right )+2 \sqrt [4]{-1} \sqrt {2} a^{3/4} c_1 \operatorname {ParabolicCylinderD}\left (\frac {i b^2+4 a^{3/2}-i a (4 c-1)}{8 a^{3/2}},\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) (b+2 a \log (x))}{a^{3/4}}\right )}{2 \sqrt {a} x \left (\operatorname {ParabolicCylinderD}\left (\frac {-i b^2-4 a^{3/2}+i a (4 c-1)}{8 a^{3/2}},-\frac {\left (\frac {1}{2}-\frac {i}{2}\right ) (b+2 a \log (x))}{a^{3/4}}\right )+c_1 \operatorname {ParabolicCylinderD}\left (\frac {i b^2-4 a^{3/2}-i a (4 c-1)}{8 a^{3/2}},\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) (b+2 a \log (x))}{a^{3/4}}\right )\right )} \\
y(x)\to -\frac {-\frac {2 \sqrt [4]{-1} \sqrt {2} \sqrt [4]{a} \operatorname {ParabolicCylinderD}\left (\frac {i b^2+4 a^{3/2}-i a (4 c-1)}{8 a^{3/2}},\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) (b+2 a \log (x))}{a^{3/4}}\right )}{\operatorname {ParabolicCylinderD}\left (\frac {i b^2-4 a^{3/2}-i a (4 c-1)}{8 a^{3/2}},\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) (b+2 a \log (x))}{a^{3/4}}\right )}+\frac {i b}{\sqrt {a}}+2 i \sqrt {a} \log (x)+1}{2 x} \\
y(x)\to -\frac {-\frac {2 \sqrt [4]{-1} \sqrt {2} \sqrt [4]{a} \operatorname {ParabolicCylinderD}\left (\frac {i b^2+4 a^{3/2}-i a (4 c-1)}{8 a^{3/2}},\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) (b+2 a \log (x))}{a^{3/4}}\right )}{\operatorname {ParabolicCylinderD}\left (\frac {i b^2-4 a^{3/2}-i a (4 c-1)}{8 a^{3/2}},\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) (b+2 a \log (x))}{a^{3/4}}\right )}+\frac {i b}{\sqrt {a}}+2 i \sqrt {a} \log (x)+1}{2 x} \\
\end{align*}