61.2.40 problem 40

Internal problem ID [12046]
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 : 40
Date solved : Monday, January 27, 2025 at 11:53:27 PM
CAS classification : [_rational, _Riccati]

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

✓ Solution by Maple

Time used: 0.005 (sec). Leaf size: 32

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

\[ y = i \tan \left (\frac {c_{1} \left (m +n \right )+i a b \,x^{m +n}}{m +n}\right ) b \,x^{m} \]

✓ Solution by Mathematica

Time used: 5.825 (sec). Leaf size: 43

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

\[ y(x)\to \sqrt {-b^2} x^m \tan \left (\frac {a \sqrt {-b^2} x^{m+n}}{m+n}+c_1\right ) \]