Internal problem ID [9659]
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]
Solve \begin {gather*} \boxed {x^{n +1} y^{\prime }-x^{2 n} a y^{2}-x^{n} b y-c \,x^{m}-d=0} \end {gather*}
✓ Solution by Maple
Time used: 0.002 (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)
\[ y \relax (x ) = \frac {\left (\left (-\sqrt {-4 a d +b^{2}+2 b n +n^{2}}\, c_{1}-b c_{1}-c_{1} n \right ) \BesselY \left (\frac {\sqrt {-4 a d +b^{2}+2 b n +n^{2}}}{m}, \frac {2 \sqrt {a c}\, x^{\frac {m}{2}}}{m}\right )+2 \sqrt {a c}\, x^{\frac {m}{2}} \BesselY \left (\frac {\sqrt {-4 a d +b^{2}+2 b n +n^{2}}+m}{m}, \frac {2 \sqrt {a c}\, x^{\frac {m}{2}}}{m}\right ) c_{1}+\left (-\sqrt {-4 a d +b^{2}+2 b n +n^{2}}-b -n \right ) \BesselJ \left (\frac {\sqrt {-4 a d +b^{2}+2 b n +n^{2}}}{m}, \frac {2 \sqrt {a c}\, x^{\frac {m}{2}}}{m}\right )+2 \BesselJ \left (\frac {\sqrt {-4 a d +b^{2}+2 b n +n^{2}}+m}{m}, \frac {2 \sqrt {a c}\, x^{\frac {m}{2}}}{m}\right ) \sqrt {a c}\, x^{\frac {m}{2}}\right ) x^{1-n}}{2 x a \left (\BesselY \left (\frac {\sqrt {-4 a d +b^{2}+2 b n +n^{2}}}{m}, \frac {2 \sqrt {a c}\, x^{\frac {m}{2}}}{m}\right ) c_{1}+\BesselJ \left (\frac {\sqrt {-4 a d +b^{2}+2 b n +n^{2}}}{m}, \frac {2 \sqrt {a c}\, x^{\frac {m}{2}}}{m}\right )\right )} \]
✓ Solution by Mathematica
Time used: 2.931 (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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