Internal problem ID [10506]
Book: Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev.
Second edition
Section: Chapter 1, section 1.2. Riccati Equation. subsection 1.2.5-2
Problem number: 21.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_Riccati]
\[ \boxed {y^{\prime } x^{2}-a^{2} x^{2} y^{2}+y x=b^{2} \ln \left (x \right )^{n}} \]
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 324
dsolve(x^2*diff(y(x),x)=a^2*x^2*y(x)^2-x*y(x)+b^2*(ln(x))^n,y(x), singsol=all)
\[ y \left (x \right ) = \frac {\frac {\ln \left (x \right )^{\frac {n}{2}+1} \sqrt {b^{2} a^{2}}\, c_{1} \operatorname {BesselY}\left (\frac {n +3}{n +2}, \frac {2 \sqrt {b^{2} a^{2}}\, \ln \left (x \right )^{\frac {n}{2}+1}}{n +2}\right )}{\left (\operatorname {BesselY}\left (\frac {1}{n +2}, \frac {2 \sqrt {b^{2} a^{2}}\, \ln \left (x \right )^{\frac {n}{2}+1}}{n +2}\right ) c_{1} +\operatorname {BesselJ}\left (\frac {1}{n +2}, \frac {2 \sqrt {b^{2} a^{2}}\, \ln \left (x \right )^{\frac {n}{2}+1}}{n +2}\right )\right ) a^{2} x}+\frac {\operatorname {BesselJ}\left (\frac {n +3}{n +2}, \frac {2 \sqrt {b^{2} a^{2}}\, \ln \left (x \right )^{\frac {n}{2}+1}}{n +2}\right ) \sqrt {b^{2} a^{2}}\, \ln \left (x \right )^{\frac {n}{2}+1}-\operatorname {BesselY}\left (\frac {1}{n +2}, \frac {2 \sqrt {b^{2} a^{2}}\, \ln \left (x \right )^{\frac {n}{2}+1}}{n +2}\right ) c_{1} -\operatorname {BesselJ}\left (\frac {1}{n +2}, \frac {2 \sqrt {b^{2} a^{2}}\, \ln \left (x \right )^{\frac {n}{2}+1}}{n +2}\right )}{\left (\operatorname {BesselY}\left (\frac {1}{n +2}, \frac {2 \sqrt {b^{2} a^{2}}\, \ln \left (x \right )^{\frac {n}{2}+1}}{n +2}\right ) c_{1} +\operatorname {BesselJ}\left (\frac {1}{n +2}, \frac {2 \sqrt {b^{2} a^{2}}\, \ln \left (x \right )^{\frac {n}{2}+1}}{n +2}\right )\right ) a^{2} x}}{\ln \left (x \right )} \]
✓ Solution by Mathematica
Time used: 45.846 (sec). Leaf size: 1769
DSolve[x^2*y'[x]==a^2*x^2*y[x]^2-x*y[x]+b^2*(Log[x])^n,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {2 b^2 \sqrt {b^{\frac {2}{n+1}} (n+2)^2} \left ((n+2)^{\frac {2 (n+1)}{n+2}} \operatorname {BesselJ}\left (\frac {n+1}{n+2},\frac {2 a \sqrt {\left (b^2 \log ^{n+1}(x)\right )^{1+\frac {1}{n+1}}}}{\sqrt {b^{\frac {2}{n+1}} (n+2)^2}}\right ) \operatorname {Gamma}\left (\frac {2 n+3}{n+2}\right ) b^{\frac {2}{n+2}}+\left (b^{\frac {2}{n+1}} (n+2)^2\right )^{\frac {n+1}{n+2}} \operatorname {BesselJ}\left (-\frac {n+1}{n+2},\frac {2 a \sqrt {\left (b^2 \log ^{n+1}(x)\right )^{1+\frac {1}{n+1}}}}{\sqrt {b^{\frac {2}{n+1}} (n+2)^2}}\right ) c_1 \operatorname {Gamma}\left (\frac {1}{n+2}\right )\right ) \log ^{n+1}(x) \sqrt {\left (b^2 \log ^{n+1}(x)\right )^{1+\frac {1}{n+1}}}}{x \left (2 a (n+2)^{\frac {2 (n+1)}{n+2}} \operatorname {BesselJ}\left (-\frac {1}{n+2},\frac {2 a \sqrt {\left (b^2 \log ^{n+1}(x)\right )^{1+\frac {1}{n+1}}}}{\sqrt {b^{\frac {2}{n+1}} (n+2)^2}}\right ) \operatorname {Gamma}\left (\frac {2 n+3}{n+2}\right ) \left (b^2 \log ^{n+1}(x)\right )^{1+\frac {1}{n+1}} b^{\frac {2}{n+2}}+a n (n+2)^{\frac {2 (n+1)}{n+2}} \operatorname {BesselJ}\left (-\frac {1}{n+2},\frac {2 a \sqrt {\left (b^2 \log ^{n+1}(x)\right )^{1+\frac {1}{n+1}}}}{\sqrt {b^{\frac {2}{n+1}} (n+2)^2}}\right ) \operatorname {Gamma}\left (\frac {2 n+3}{n+2}\right ) \left (b^2 \log ^{n+1}(x)\right )^{1+\frac {1}{n+1}} b^{\frac {2}{n+2}}-2 a (n+2)^{\frac {2 (n+1)}{n+2}} \operatorname {BesselJ}\left (\frac {2 n+3}{n+2},\frac {2 a \sqrt {\left (b^2 \log ^{n+1}(x)\right )^{1+\frac {1}{n+1}}}}{\sqrt {b^{\frac {2}{n+1}} (n+2)^2}}\right ) \operatorname {Gamma}\left (\frac {2 n+3}{n+2}\right ) \left (b^2 \log ^{n+1}(x)\right )^{1+\frac {1}{n+1}} b^{\frac {2}{n+2}}-a n (n+2)^{\frac {2 (n+1)}{n+2}} \operatorname {BesselJ}\left (\frac {2 n+3}{n+2},\frac {2 a \sqrt {\left (b^2 \log ^{n+1}(x)\right )^{1+\frac {1}{n+1}}}}{\sqrt {b^{\frac {2}{n+1}} (n+2)^2}}\right ) \operatorname {Gamma}\left (\frac {2 n+3}{n+2}\right ) \left (b^2 \log ^{n+1}(x)\right )^{1+\frac {1}{n+1}} b^{\frac {2}{n+2}}+n (n+2)^{\frac {2 (n+1)}{n+2}} \sqrt {b^{\frac {2}{n+1}} (n+2)^2} \operatorname {BesselJ}\left (\frac {n+1}{n+2},\frac {2 a \sqrt {\left (b^2 \log ^{n+1}(x)\right )^{1+\frac {1}{n+1}}}}{\sqrt {b^{\frac {2}{n+1}} (n+2)^2}}\right ) \operatorname {Gamma}\left (\frac {2 n+3}{n+2}\right ) \sqrt {\left (b^2 \log ^{n+1}(x)\right )^{1+\frac {1}{n+1}}} b^{\frac {2}{n+2}}+(n+2)^{\frac {2 (n+1)}{n+2}} \sqrt {b^{\frac {2}{n+1}} (n+2)^2} \operatorname {BesselJ}\left (\frac {n+1}{n+2},\frac {2 a \sqrt {\left (b^2 \log ^{n+1}(x)\right )^{1+\frac {1}{n+1}}}}{\sqrt {b^{\frac {2}{n+1}} (n+2)^2}}\right ) \operatorname {Gamma}\left (\frac {2 n+3}{n+2}\right ) \sqrt {\left (b^2 \log ^{n+1}(x)\right )^{1+\frac {1}{n+1}}} b^{\frac {2}{n+2}}-a (n+2) \left (b^{\frac {2}{n+1}} (n+2)^2\right )^{\frac {n+1}{n+2}} \operatorname {BesselJ}\left (\frac {1}{n+2},\frac {2 a \sqrt {\left (b^2 \log ^{n+1}(x)\right )^{1+\frac {1}{n+1}}}}{\sqrt {b^{\frac {2}{n+1}} (n+2)^2}}\right ) c_1 \operatorname {Gamma}\left (\frac {1}{n+2}\right ) \left (b^2 \log ^{n+1}(x)\right )^{1+\frac {1}{n+1}}+a (n+2) \left (b^{\frac {2}{n+1}} (n+2)^2\right )^{\frac {n+1}{n+2}} \operatorname {BesselJ}\left (-\frac {2 n+3}{n+2},\frac {2 a \sqrt {\left (b^2 \log ^{n+1}(x)\right )^{1+\frac {1}{n+1}}}}{\sqrt {b^{\frac {2}{n+1}} (n+2)^2}}\right ) c_1 \operatorname {Gamma}\left (\frac {1}{n+2}\right ) \left (b^2 \log ^{n+1}(x)\right )^{1+\frac {1}{n+1}}+n \left (b^{\frac {2}{n+1}} (n+2)^2\right )^{\frac {3 n+4}{2 n+4}} \operatorname {BesselJ}\left (-\frac {n+1}{n+2},\frac {2 a \sqrt {\left (b^2 \log ^{n+1}(x)\right )^{1+\frac {1}{n+1}}}}{\sqrt {b^{\frac {2}{n+1}} (n+2)^2}}\right ) c_1 \operatorname {Gamma}\left (\frac {1}{n+2}\right ) \sqrt {\left (b^2 \log ^{n+1}(x)\right )^{1+\frac {1}{n+1}}}+\left (b^{\frac {2}{n+1}} (n+2)^2\right )^{\frac {3 n+4}{2 n+4}} \operatorname {BesselJ}\left (-\frac {n+1}{n+2},\frac {2 a \sqrt {\left (b^2 \log ^{n+1}(x)\right )^{1+\frac {1}{n+1}}}}{\sqrt {b^{\frac {2}{n+1}} (n+2)^2}}\right ) c_1 \operatorname {Gamma}\left (\frac {1}{n+2}\right ) \sqrt {\left (b^2 \log ^{n+1}(x)\right )^{1+\frac {1}{n+1}}}\right )} y(x)\to \frac {2 b^2 \sqrt {(n+2)^2 b^{\frac {2}{n+1}}} \log ^{n+1}(x) \sqrt {\left (b^2 \log ^{n+1}(x)\right )^{\frac {1}{n+1}+1}} \operatorname {BesselJ}\left (-\frac {n+1}{n+2},\frac {2 a \sqrt {\left (b^2 \log ^{n+1}(x)\right )^{1+\frac {1}{n+1}}}}{\sqrt {b^{\frac {2}{n+1}} (n+2)^2}}\right )}{x \left (-a (n+2) \left (b^2 \log ^{n+1}(x)\right )^{\frac {1}{n+1}+1} \operatorname {BesselJ}\left (\frac {1}{n+2},\frac {2 a \sqrt {\left (b^2 \log ^{n+1}(x)\right )^{1+\frac {1}{n+1}}}}{\sqrt {b^{\frac {2}{n+1}} (n+2)^2}}\right )+a (n+2) \left (b^2 \log ^{n+1}(x)\right )^{\frac {1}{n+1}+1} \operatorname {BesselJ}\left (-\frac {2 n+3}{n+2},\frac {2 a \sqrt {\left (b^2 \log ^{n+1}(x)\right )^{1+\frac {1}{n+1}}}}{\sqrt {b^{\frac {2}{n+1}} (n+2)^2}}\right )+(n+1) \sqrt {(n+2)^2 b^{\frac {2}{n+1}}} \sqrt {\left (b^2 \log ^{n+1}(x)\right )^{\frac {1}{n+1}+1}} \operatorname {BesselJ}\left (-\frac {n+1}{n+2},\frac {2 a \sqrt {\left (b^2 \log ^{n+1}(x)\right )^{1+\frac {1}{n+1}}}}{\sqrt {b^{\frac {2}{n+1}} (n+2)^2}}\right )\right )} \end{align*}