Internal problem ID [10488]
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]
\[ \boxed {y^{\prime } x -a y^{2}=b \ln \left (x \right )^{k}+c \ln \left (x \right )^{2 k +2}} \]
✓ Solution by Maple
Time used: 0.016 (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 \left (x \right ) = \frac {\left (\left (-i \sqrt {a}\, \ln \left (x \right )^{k +2} \sqrt {c}\, c_{1} k^{2}-4 i \sqrt {a}\, \ln \left (x \right )^{k +2} \sqrt {c}\, c_{1} k -3 i \sqrt {a}\, \ln \left (x \right )^{k +2} \sqrt {c}\, c_{1} +\ln \left (x \right )^{k +2} c_{1} a b k +\ln \left (x \right )^{k +2} c_{1} a b \right ) \operatorname {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 {a}\, \sqrt {c}\, \ln \left (x \right )^{k +2}}{k +2}\right )+\left (i \sqrt {a}\, \ln \left (x \right )^{k +2} \sqrt {c}\, c_{1} k^{2}+4 i \sqrt {a}\, \ln \left (x \right )^{k +2} \sqrt {c}\, c_{1} k +3 i \sqrt {a}\, \ln \left (x \right )^{k +2} \sqrt {c}\, c_{1} -c_{1} k^{2}-4 c_{1} k -3 c_{1} \right ) \operatorname {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 {a}\, \sqrt {c}\, \ln \left (x \right )^{k +2}}{k +2}\right )\right ) \ln \left (x \right )+\left (-i \sqrt {a}\, \ln \left (x \right )^{k +2} \sqrt {c}\, k^{2}-4 i \sqrt {a}\, \ln \left (x \right )^{k +2} \sqrt {c}\, k -3 i \sqrt {a}\, \sqrt {c}\, \ln \left (x \right )^{k +2}+\ln \left (x \right )^{k +2} a b k +3 \ln \left (x \right )^{k +2} a b \right ) \operatorname {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 {a}\, \sqrt {c}\, \ln \left (x \right )^{k +2}}{k +2}\right )+\left (i \sqrt {a}\, \ln \left (x \right )^{k +2} \sqrt {c}\, k^{2}+4 i \sqrt {a}\, \ln \left (x \right )^{k +2} \sqrt {c}\, k +3 i \sqrt {a}\, \sqrt {c}\, \ln \left (x \right )^{k +2}\right ) \operatorname {hypergeom}\left (\left [\frac {i \sqrt {a}\, b +k \sqrt {c}+\sqrt {c}}{2 \sqrt {c}\, \left (k +2\right )}\right ], \left [\frac {1+k}{k +2}\right ], \frac {2 i \sqrt {a}\, \sqrt {c}\, \ln \left (x \right )^{k +2}}{k +2}\right )}{\left (\operatorname {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 {a}\, \sqrt {c}\, \ln \left (x \right )^{k +2}}{k +2}\right ) \ln \left (x \right ) c_{1} +\operatorname {hypergeom}\left (\left [\frac {i \sqrt {a}\, b +k \sqrt {c}+\sqrt {c}}{2 \sqrt {c}\, \left (k +2\right )}\right ], \left [\frac {1+k}{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 (1+k \right ) \ln \left (x \right )} \]
✓ Solution by Mathematica
Time used: 3.775 (sec). Leaf size: 807
DSolve[x*y'[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} \end{align*}