Internal problem ID [10381]
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: 51.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_rational, _Riccati]
\[ \boxed {x^{2} y^{\prime }-a \,x^{2} y^{2}-y b x=c \,x^{n}+s} \]
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 263
dsolve(x^2*diff(y(x),x)=a*x^2*y(x)^2+b*x*y(x)+c*x^n+s,y(x), singsol=all)
\[ y \left (x \right ) = \frac {2 \left (\operatorname {BesselY}\left (\frac {\sqrt {-4 a s +b^{2}+2 b +1}}{n}+1, \frac {2 \sqrt {a c}\, x^{\frac {n}{2}}}{n}\right ) c_{1} +\operatorname {BesselJ}\left (\frac {\sqrt {-4 a s +b^{2}+2 b +1}}{n}+1, \frac {2 \sqrt {a c}\, x^{\frac {n}{2}}}{n}\right )\right ) \sqrt {a c}\, x^{\frac {n}{2}}-\left (\operatorname {BesselY}\left (\frac {\sqrt {-4 a s +b^{2}+2 b +1}}{n}, \frac {2 \sqrt {a c}\, x^{\frac {n}{2}}}{n}\right ) c_{1} +\operatorname {BesselJ}\left (\frac {\sqrt {-4 a s +b^{2}+2 b +1}}{n}, \frac {2 \sqrt {a c}\, x^{\frac {n}{2}}}{n}\right )\right ) \left (\sqrt {-4 a s +b^{2}+2 b +1}+b +1\right )}{2 x a \left (\operatorname {BesselY}\left (\frac {\sqrt {-4 a s +b^{2}+2 b +1}}{n}, \frac {2 \sqrt {a c}\, x^{\frac {n}{2}}}{n}\right ) c_{1} +\operatorname {BesselJ}\left (\frac {\sqrt {-4 a s +b^{2}+2 b +1}}{n}, \frac {2 \sqrt {a c}\, x^{\frac {n}{2}}}{n}\right )\right )} \]
✓ Solution by Mathematica
Time used: 2.637 (sec). Leaf size: 2281
DSolve[x^2*y'[x]==a*x^2*y[x]^2+b*x*y[x]+c*x^n+s,y[x],x,IncludeSingularSolutions -> True]
Too large to display