Internal problem ID [9621]
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: 34.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_rational, _Riccati]
Solve \begin {gather*} \boxed {y^{\prime } x -a y^{2}-b y-c \,x^{2 b}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.009 (sec). Leaf size: 38
dsolve(x*diff(y(x),x)=a*y(x)^2+b*y(x)+c*x^(2*b),y(x), singsol=all)
\[ y \relax (x ) = \frac {\tan \left (\frac {x^{b} \sqrt {c}\, \sqrt {a}-b c_{1}}{b}\right ) \sqrt {c}\, x^{b}}{\sqrt {a}} \]
✓ Solution by Mathematica
Time used: 0.546 (sec). Leaf size: 139
DSolve[x*y'[x]==a*y[x]^2+b*y[x]+c*x^(2*b),y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {\sqrt {c} x^b \left (-\cos \left (\frac {\sqrt {a} \sqrt {c} x^b}{b}\right )+c_1 \sin \left (\frac {\sqrt {a} \sqrt {c} x^b}{b}\right )\right )}{\sqrt {a} \left (\sin \left (\frac {\sqrt {a} \sqrt {c} x^b}{b}\right )+c_1 \cos \left (\frac {\sqrt {a} \sqrt {c} x^b}{b}\right )\right )} \\ y(x)\to \frac {\sqrt {c} x^b \tan \left (\frac {\sqrt {a} \sqrt {c} x^b}{b}\right )}{\sqrt {a}} \\ \end{align*}