Internal problem ID [9634]
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]
\[ \boxed {x^{2} y^{\prime }-a \,x^{2} y^{2}-b x y-c \,x^{n}-s=0} \]
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 299
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)
\[ y \left (x \right ) = \frac {\left (-\sqrt {-4 a s +b^{2}+2 b +1}\, c_{1} -c_{1} b -c_{1} \right ) \operatorname {BesselY}\left (\frac {\sqrt {-4 a s +b^{2}+2 b +1}}{n}, \frac {2 \sqrt {a c}\, x^{\frac {n}{2}}}{n}\right )+2 \sqrt {a c}\, \operatorname {BesselY}\left (\frac {\sqrt {-4 a s +b^{2}+2 b +1}+n}{n}, \frac {2 \sqrt {a c}\, x^{\frac {n}{2}}}{n}\right ) x^{\frac {n}{2}} c_{1} +\left (-\sqrt {-4 a s +b^{2}+2 b +1}-b -1\right ) \operatorname {BesselJ}\left (\frac {\sqrt {-4 a s +b^{2}+2 b +1}}{n}, \frac {2 \sqrt {a c}\, x^{\frac {n}{2}}}{n}\right )+2 \operatorname {BesselJ}\left (\frac {\sqrt {-4 a s +b^{2}+2 b +1}+n}{n}, \frac {2 \sqrt {a c}\, x^{\frac {n}{2}}}{n}\right ) \sqrt {a c}\, x^{\frac {n}{2}}}{2 x a \left (\operatorname {BesselY}\left (\frac {\sqrt {-4 a s +b^{2}+2 b +1}}{n}, \frac {2 \sqrt {a c}\, x^{\frac {n}{2}}}{n}\right ) c_{1} +\operatorname {BesselJ}\left (\frac {\sqrt {-4 a s +b^{2}+2 b +1}}{n}, \frac {2 \sqrt {a c}\, x^{\frac {n}{2}}}{n}\right )\right )} \]
✓ Solution by Mathematica
Time used: 1.41 (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]
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