61.2.34 problem 34

Internal problem ID [12040]
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
Date solved : Monday, January 27, 2025 at 11:53:06 PM
CAS classification : [_rational, _Riccati]

\begin{align*} x y^{\prime }&=a y^{2}+b y+c \,x^{2 b} \end{align*}

✓ Solution by Maple

Time used: 0.044 (sec). Leaf size: 34

dsolve(x*diff(y(x),x)=a*y(x)^2+b*y(x)+c*x^(2*b),y(x), singsol=all)
 

\[ y = \frac {\tan \left (\frac {x^{b} \sqrt {a}\, \sqrt {c}-c_{1} b}{b}\right ) \sqrt {c}\, x^{b}}{\sqrt {a}} \]

✓ Solution by Mathematica

Time used: 0.344 (sec). Leaf size: 139

DSolve[x*D[y[x],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*}