Internal problem ID [9749]
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]
\[ \boxed {x^{2} y^{\prime }-a^{2} x^{2} y^{2}+x y-b^{2} \ln \left (x \right )^{n}=0} \]
✓ 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: 0.0 (sec). Leaf size: 0
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]
Not solved