61.8.12 problem 21

Internal problem ID [12172]
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
Date solved : Tuesday, January 28, 2025 at 12:47:10 AM
CAS classification : [_Riccati]

\begin{align*} x^{2} y^{\prime }&=a^{2} x^{2} y^{2}-y x +b^{2} \ln \left (x \right )^{n} \end{align*}

✓ Solution by Maple

Time used: 0.003 (sec). Leaf size: 251

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)
 

\[ y = \frac {\sqrt {b^{2} a^{2}}\, \operatorname {BesselY}\left (\frac {n +3}{n +2}, \frac {2 \sqrt {b^{2} a^{2}}\, \ln \left (x \right )^{\frac {n}{2}+1}}{n +2}\right ) \ln \left (x \right )^{\frac {n}{2}+1} c_{1} +\operatorname {BesselJ}\left (\frac {n +3}{n +2}, \frac {2 \sqrt {b^{2} a^{2}}\, \ln \left (x \right )^{\frac {n}{2}+1}}{n +2}\right ) \sqrt {b^{2} a^{2}}\, \ln \left (x \right )^{\frac {n}{2}+1}-\operatorname {BesselY}\left (\frac {1}{n +2}, \frac {2 \sqrt {b^{2} a^{2}}\, \ln \left (x \right )^{\frac {n}{2}+1}}{n +2}\right ) c_{1} -\operatorname {BesselJ}\left (\frac {1}{n +2}, \frac {2 \sqrt {b^{2} a^{2}}\, \ln \left (x \right )^{\frac {n}{2}+1}}{n +2}\right )}{\left (\operatorname {BesselY}\left (\frac {1}{n +2}, \frac {2 \sqrt {b^{2} a^{2}}\, \ln \left (x \right )^{\frac {n}{2}+1}}{n +2}\right ) c_{1} +\operatorname {BesselJ}\left (\frac {1}{n +2}, \frac {2 \sqrt {b^{2} a^{2}}\, \ln \left (x \right )^{\frac {n}{2}+1}}{n +2}\right )\right ) a^{2} x \ln \left (x \right )} \]

✓ Solution by Mathematica

Time used: 42.129 (sec). Leaf size: 1769

DSolve[x^2*D[y[x],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*}