61.2.43 problem 43

Internal problem ID [12049]
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 : 43
Date solved : Monday, January 27, 2025 at 11:53:35 PM
CAS classification : [_rational, _Riccati]

\begin{align*} x y^{\prime }&=a \,x^{2 n} y^{2}+\left (b \,x^{n}-n \right ) y+c \end{align*}

✓ Solution by Maple

Time used: 0.030 (sec). Leaf size: 72

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

\[ y = \frac {\left (\sqrt {4 a \,b^{2} c -b^{4}}\, \tan \left (\frac {\sqrt {4 a \,b^{2} c -b^{4}}\, \left (b \,x^{n}+c_{1} n \right )}{2 b^{2} n}\right )-b^{2}\right ) x^{-n}}{2 a b} \]

✓ Solution by Mathematica

Time used: 0.771 (sec). Leaf size: 118

DSolve[x*D[y[x],x]==a*x^(2*n)*y[x]^2+(b*x^n-n)*y[x]+c,y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to \frac {x^{-n} \left (-b+\frac {\sqrt {b^2-4 a c} \left (-e^{\frac {x^n \sqrt {b^2-4 a c}}{n}}+c_1\right )}{e^{\frac {x^n \sqrt {b^2-4 a c}}{n}}+c_1}\right )}{2 a} \\ y(x)\to \frac {x^{-n} \left (\sqrt {b^2-4 a c}-b\right )}{2 a} \\ \end{align*}