Internal problem ID [9753]
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]
Solve \begin {gather*} \boxed {x^{2} y^{\prime }-a^{2} x^{2} y^{2}+y x -b^{2} \ln \relax (x )^{n}=0} \end {gather*}
✓ 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)
\[ y \relax (x ) = \frac {\frac {\ln \relax (x )^{1+\frac {n}{2}} \sqrt {b^{2} a^{2}}\, c_{1} \BesselY \left (\frac {n +3}{2+n}, \frac {2 \sqrt {b^{2} a^{2}}\, \ln \relax (x )^{1+\frac {n}{2}}}{2+n}\right )}{\left (\BesselY \left (\frac {1}{2+n}, \frac {2 \sqrt {b^{2} a^{2}}\, \ln \relax (x )^{1+\frac {n}{2}}}{2+n}\right ) c_{1}+\BesselJ \left (\frac {1}{2+n}, \frac {2 \sqrt {b^{2} a^{2}}\, \ln \relax (x )^{1+\frac {n}{2}}}{2+n}\right )\right ) a^{2} x}+\frac {\BesselJ \left (\frac {n +3}{2+n}, \frac {2 \sqrt {b^{2} a^{2}}\, \ln \relax (x )^{1+\frac {n}{2}}}{2+n}\right ) \sqrt {b^{2} a^{2}}\, \ln \relax (x )^{1+\frac {n}{2}}-\BesselY \left (\frac {1}{2+n}, \frac {2 \sqrt {b^{2} a^{2}}\, \ln \relax (x )^{1+\frac {n}{2}}}{2+n}\right ) c_{1}-\BesselJ \left (\frac {1}{2+n}, \frac {2 \sqrt {b^{2} a^{2}}\, \ln \relax (x )^{1+\frac {n}{2}}}{2+n}\right )}{\left (\BesselY \left (\frac {1}{2+n}, \frac {2 \sqrt {b^{2} a^{2}}\, \ln \relax (x )^{1+\frac {n}{2}}}{2+n}\right ) c_{1}+\BesselJ \left (\frac {1}{2+n}, \frac {2 \sqrt {b^{2} a^{2}}\, \ln \relax (x )^{1+\frac {n}{2}}}{2+n}\right )\right ) a^{2} x}}{\ln \relax (x )} \]
✗ 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