61.2.41 problem 41

Internal problem ID [12047]
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 : 41
Date solved : Monday, January 27, 2025 at 11:53:29 PM
CAS classification : [_rational, _Riccati]

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

✓ Solution by Maple

Time used: 0.004 (sec). Leaf size: 34

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

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

✓ Solution by Mathematica

Time used: 0.488 (sec). Leaf size: 28

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

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