Internal problem ID [10402]
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]
\[ \boxed {x^{n +1} y^{\prime }-x^{2 n} y^{2} a -y x^{n} b=c \,x^{m}+d} \]
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 284
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)
\[ y \left (x \right ) = \frac {\left (\sqrt {a c}\, \left (\operatorname {BesselY}\left (\frac {\sqrt {-4 a d +b^{2}+2 b n +n^{2}}}{m}+1, \frac {2 \sqrt {a c}\, x^{\frac {m}{2}}}{m}\right ) c_{1} +\operatorname {BesselJ}\left (\frac {\sqrt {-4 a d +b^{2}+2 b n +n^{2}}}{m}+1, \frac {2 \sqrt {a c}\, x^{\frac {m}{2}}}{m}\right )\right ) x^{\frac {m}{2}}-\frac {\left (\operatorname {BesselY}\left (\frac {\sqrt {-4 a d +b^{2}+2 b n +n^{2}}}{m}, \frac {2 \sqrt {a c}\, x^{\frac {m}{2}}}{m}\right ) c_{1} +\operatorname {BesselJ}\left (\frac {\sqrt {-4 a d +b^{2}+2 b n +n^{2}}}{m}, \frac {2 \sqrt {a c}\, x^{\frac {m}{2}}}{m}\right )\right ) \left (\sqrt {-4 a d +b^{2}+2 b n +n^{2}}+b +n \right )}{2}\right ) x^{-n}}{a \left (\operatorname {BesselY}\left (\frac {\sqrt {-4 a d +b^{2}+2 b n +n^{2}}}{m}, \frac {2 \sqrt {a c}\, x^{\frac {m}{2}}}{m}\right ) c_{1} +\operatorname {BesselJ}\left (\frac {\sqrt {-4 a d +b^{2}+2 b n +n^{2}}}{m}, \frac {2 \sqrt {a c}\, x^{\frac {m}{2}}}{m}\right )\right )} \]
✓ Solution by Mathematica
Time used: 3.153 (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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