Internal problem ID [10482]
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.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_Riccati]
\[ \boxed {x^{2} y^{\prime }-x^{2} y^{2}=a \ln \left (x \right )^{2}+b \ln \left (x \right )+c} \]
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 477
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 \left (x \right ) = \frac {\left (-\frac {b \ln \left (x \right ) \left (b \ln \left (x \right )-4 c +1\right ) a^{\frac {5}{2}}}{12}-\frac {\left (b \ln \left (x \right )-c +\frac {1}{4}\right ) b^{2} a^{\frac {3}{2}}}{12}+\frac {\left (c -\frac {1}{4}\right ) \ln \left (x \right )^{2} a^{\frac {7}{2}}}{3}-\frac {\sqrt {a}\, b^{4}}{48}-i \left (a \ln \left (x \right )+\frac {b}{2}\right )^{2} a^{2}\right ) c_{1} \operatorname {hypergeom}\left (\left [\frac {28 a^{\frac {3}{2}}+i \left (4 c -1\right ) a -i b^{2}}{16 a^{\frac {3}{2}}}\right ], \left [\frac {5}{2}\right ], \frac {i \left (2 a \ln \left (x \right )+b \right )^{2}}{4 a^{\frac {3}{2}}}\right )+c_{1} \left (-\frac {a^{\frac {5}{2}} b}{4}+\left (-1-\frac {\ln \left (x \right )}{2}\right ) a^{\frac {7}{2}}+i \left (a \ln \left (x \right )+\frac {b}{2}\right )^{2} a^{2}\right ) \operatorname {hypergeom}\left (\left [\frac {12 a^{\frac {3}{2}}+i \left (4 c -1\right ) a -i b^{2}}{16 a^{\frac {3}{2}}}\right ], \left [\frac {3}{2}\right ], \frac {i \left (2 a \ln \left (x \right )+b \right )^{2}}{4 a^{\frac {3}{2}}}\right )+\frac {\left (\left (c -\frac {1}{4}\right ) \ln \left (x \right ) a^{\frac {5}{2}}-\frac {b \left (b \ln \left (x \right )-2 c +\frac {1}{2}\right ) a^{\frac {3}{2}}}{4}-\frac {\sqrt {a}\, b^{3}}{8}-i \left (a \ln \left (x \right )+\frac {b}{2}\right ) a^{2}\right ) \operatorname {hypergeom}\left (\left [\frac {20 a^{\frac {3}{2}}+i \left (4 c -1\right ) a -i b^{2}}{16 a^{\frac {3}{2}}}\right ], \left [\frac {3}{2}\right ], \frac {i \left (2 a \ln \left (x \right )+b \right )^{2}}{4 a^{\frac {3}{2}}}\right )}{2}+\frac {\left (i a^{3} \ln \left (x \right )+\frac {i a^{2} b}{2}-\frac {a^{\frac {5}{2}}}{2}\right ) \operatorname {hypergeom}\left (\left [\frac {4 a^{\frac {3}{2}}+i \left (4 c -1\right ) a -i b^{2}}{16 a^{\frac {3}{2}}}\right ], \left [\frac {1}{2}\right ], \frac {i \left (2 a \ln \left (x \right )+b \right )^{2}}{4 a^{\frac {3}{2}}}\right )}{2}}{a^{\frac {5}{2}} x \left (c_{1} \left (a \ln \left (x \right )+\frac {b}{2}\right ) \operatorname {hypergeom}\left (\left [\frac {12 a^{\frac {3}{2}}+i \left (4 c -1\right ) a -i b^{2}}{16 a^{\frac {3}{2}}}\right ], \left [\frac {3}{2}\right ], \frac {i \left (2 a \ln \left (x \right )+b \right )^{2}}{4 a^{\frac {3}{2}}}\right )+\frac {\operatorname {hypergeom}\left (\left [\frac {4 a^{\frac {3}{2}}+i \left (4 c -1\right ) a -i b^{2}}{16 a^{\frac {3}{2}}}\right ], \left [\frac {1}{2}\right ], \frac {i \left (2 a \ln \left (x \right )+b \right )^{2}}{4 a^{\frac {3}{2}}}\right )}{2}\right )} \]
✓ Solution by Mathematica
Time used: 1.151 (sec). Leaf size: 868
DSolve[x^2*y'[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*}