Internal problem ID [9735]
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.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_Riccati]
Solve \begin {gather*} \boxed {y^{\prime } x -a y^{2}-b \ln \relax (x )^{k}-c \ln \relax (x )^{2 k +2}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 660
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 \relax (x ) = \frac {\left (\left (-i \sqrt {c}\, \ln \relax (x )^{k +2} \sqrt {a}\, c_{1} k^{2}-4 i \sqrt {c}\, \ln \relax (x )^{k +2} \sqrt {a}\, c_{1} k -3 i \sqrt {c}\, \ln \relax (x )^{k +2} \sqrt {a}\, c_{1}+\ln \relax (x )^{k +2} c_{1} a b k +\ln \relax (x )^{k +2} c_{1} a b \right ) \hypergeom \left (\left [\frac {i \sqrt {a}\, b +3 k \sqrt {c}+7 \sqrt {c}}{2 \sqrt {c}\, \left (k +2\right )}\right ], \left [\frac {2 k +5}{k +2}\right ], \frac {2 i \sqrt {c}\, \sqrt {a}\, \ln \relax (x )^{k +2}}{k +2}\right )+\left (i \sqrt {c}\, \ln \relax (x )^{k +2} \sqrt {a}\, c_{1} k^{2}+4 i \sqrt {c}\, \ln \relax (x )^{k +2} \sqrt {a}\, c_{1} k +3 i \sqrt {c}\, \ln \relax (x )^{k +2} \sqrt {a}\, c_{1}-c_{1} k^{2}-4 c_{1} k -3 c_{1}\right ) \hypergeom \left (\left [\frac {i \sqrt {a}\, b +k \sqrt {c}+3 \sqrt {c}}{2 \sqrt {c}\, \left (k +2\right )}\right ], \left [\frac {k +3}{k +2}\right ], \frac {2 i \sqrt {c}\, \sqrt {a}\, \ln \relax (x )^{k +2}}{k +2}\right )\right ) \ln \relax (x )+\left (-i \sqrt {c}\, \ln \relax (x )^{k +2} \sqrt {a}\, k^{2}-4 i \sqrt {c}\, \ln \relax (x )^{k +2} \sqrt {a}\, k -3 i \sqrt {c}\, \sqrt {a}\, \ln \relax (x )^{k +2}+\ln \relax (x )^{k +2} a b k +3 \ln \relax (x )^{k +2} a b \right ) \hypergeom \left (\left [\frac {i \sqrt {a}\, b +3 k \sqrt {c}+5 \sqrt {c}}{2 \sqrt {c}\, \left (k +2\right )}\right ], \left [\frac {2 k +3}{k +2}\right ], \frac {2 i \sqrt {c}\, \sqrt {a}\, \ln \relax (x )^{k +2}}{k +2}\right )+\left (i \sqrt {c}\, \ln \relax (x )^{k +2} \sqrt {a}\, k^{2}+4 i \sqrt {c}\, \ln \relax (x )^{k +2} \sqrt {a}\, k +3 i \sqrt {c}\, \sqrt {a}\, \ln \relax (x )^{k +2}\right ) \hypergeom \left (\left [\frac {i \sqrt {a}\, b +k \sqrt {c}+\sqrt {c}}{2 \sqrt {c}\, \left (k +2\right )}\right ], \left [\frac {k +1}{k +2}\right ], \frac {2 i \sqrt {c}\, \sqrt {a}\, \ln \relax (x )^{k +2}}{k +2}\right )}{\left (\ln \relax (x ) \hypergeom \left (\left [\frac {i \sqrt {a}\, b +k \sqrt {c}+3 \sqrt {c}}{2 \sqrt {c}\, \left (k +2\right )}\right ], \left [\frac {k +3}{k +2}\right ], \frac {2 i \sqrt {c}\, \sqrt {a}\, \ln \relax (x )^{k +2}}{k +2}\right ) c_{1}+\hypergeom \left (\left [\frac {i \sqrt {a}\, b +k \sqrt {c}+\sqrt {c}}{2 \sqrt {c}\, \left (k +2\right )}\right ], \left [\frac {k +1}{k +2}\right ], \frac {2 i \sqrt {c}\, \sqrt {a}\, \ln \relax (x )^{k +2}}{k +2}\right )\right ) a \left (k +3\right ) \left (k +1\right ) \ln \relax (x )} \]
✗ Solution by Mathematica
Time used: 0.0 (sec). Leaf size: 0
DSolve[x*y'[x]==a*y[x]^2+b*(Log[x])^k+c*(Log[x])^(2*k+2),y[x],x,IncludeSingularSolutions -> True]
Not solved