61.19.7 problem 7

Internal problem ID [12276]
Book : Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev. Second edition
Section : Chapter 1, section 1.2. Riccati Equation. subsection 1.2.8-1. Equations containing arbitrary functions (but not containing their derivatives).
Problem number : 7
Date solved : Tuesday, January 28, 2025 at 02:02:59 AM
CAS classification : [_Riccati]

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

✓ Solution by Maple

Time used: 0.115 (sec). Leaf size: 30

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

\[ y = -\tan \left (-\sqrt {a}\, \left (\int f x^{n -1}d x \right )+c_{1} \right ) \sqrt {a}\, x^{n} \]

✓ Solution by Mathematica

Time used: 0.390 (sec). Leaf size: 41

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

\[ y(x)\to \sqrt {a} x^n \tan \left (\sqrt {a} \int _1^xf(K[1]) K[1]^{n-1}dK[1]+c_1\right ) \]