Internal problem ID [9655]
Book: Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev. Second
edition
Section: Chapter 1, section 1.2. Riccati Equation. 1.2.2. Equations Containing Power Functions
Problem number: 72.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_Riccati]
\[ \boxed {x^{1+n} y^{\prime }-x^{2 n} y^{2} a -x^{n} y b -c \,x^{m}-d=0} \]
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 333
dsolve(x^(n+1)*diff(y(x),x)=a*x^(2*n)*y(x)^2+b*x^n*y(x)+c*x^m+d,y(x), singsol=all)
\[ y \left (x \right ) = \frac {\left (\left (-\sqrt {-4 a d +b^{2}+2 b n +n^{2}}\, c_{1} -c_{1} b -c_{1} n \right ) \operatorname {BesselY}\left (\frac {\sqrt {-4 a d +b^{2}+2 b n +n^{2}}}{m}, \frac {2 \sqrt {a c}\, x^{\frac {m}{2}}}{m}\right )+2 \sqrt {a c}\, \operatorname {BesselY}\left (\frac {\sqrt {-4 a d +b^{2}+2 b n +n^{2}}+m}{m}, \frac {2 \sqrt {a c}\, x^{\frac {m}{2}}}{m}\right ) x^{\frac {m}{2}} c_{1} +\left (-\sqrt {-4 a d +b^{2}+2 b n +n^{2}}-b -n \right ) \operatorname {BesselJ}\left (\frac {\sqrt {-4 a d +b^{2}+2 b n +n^{2}}}{m}, \frac {2 \sqrt {a c}\, x^{\frac {m}{2}}}{m}\right )+2 \operatorname {BesselJ}\left (\frac {\sqrt {-4 a d +b^{2}+2 b n +n^{2}}+m}{m}, \frac {2 \sqrt {a c}\, x^{\frac {m}{2}}}{m}\right ) \sqrt {a c}\, x^{\frac {m}{2}}\right ) x^{-n +1}}{2 x a \left (\operatorname {BesselY}\left (\frac {\sqrt {-4 a d +b^{2}+2 b n +n^{2}}}{m}, \frac {2 \sqrt {a c}\, x^{\frac {m}{2}}}{m}\right ) c_{1} +\operatorname {BesselJ}\left (\frac {\sqrt {-4 a d +b^{2}+2 b n +n^{2}}}{m}, \frac {2 \sqrt {a c}\, x^{\frac {m}{2}}}{m}\right )\right )} \]
✓ Solution by Mathematica
Time used: 1.708 (sec). Leaf size: 2576
DSolve[x^(n+1)*y'[x]==a*x^(2*n)*y[x]^2+b*x^n*y[x]+c*x^m+d,y[x],x,IncludeSingularSolutions -> True]
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