61.19.6 problem 6

Internal problem ID [12275]
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 : 6
Date solved : Tuesday, January 28, 2025 at 02:02:56 AM
CAS classification : [_Riccati]

\begin{align*} y^{\prime }&=-\left (n +1\right ) x^{n} y^{2}+x^{n +1} f \left (x \right ) y-f \left (x \right ) \end{align*}

✓ Solution by Maple

Time used: 0.014 (sec). Leaf size: 169

dsolve(diff(y(x),x)=-(n+1)*x^n*y(x)^2+x^(n+1)*f(x)*y(x)-f(x),y(x), singsol=all)
 

\[ y = \frac {x^{-n -1} \left (x^{n +1} {\mathrm e}^{\int \frac {x^{n +1} f x -2 n -2}{x}d x}+\left (\int x^{n} {\mathrm e}^{\int x^{n +1} fd x +\left (-2 n -2\right ) \left (\int \frac {1}{x}d x \right )}d x \right ) n +\int x^{n} {\mathrm e}^{\int x^{n +1} fd x +\left (-2 n -2\right ) \left (\int \frac {1}{x}d x \right )}d x -c_{1} \right )}{\left (\int x^{n} {\mathrm e}^{\int x^{n +1} fd x +\left (-2 n -2\right ) \left (\int \frac {1}{x}d x \right )}d x \right ) n +\int x^{n} {\mathrm e}^{\int x^{n +1} fd x +\left (-2 n -2\right ) \left (\int \frac {1}{x}d x \right )}d x -c_{1}} \]

✗ Solution by Mathematica

Time used: 0.000 (sec). Leaf size: 0

DSolve[D[y[x],x]==-(n+1)*x^n*y[x]^2+x^(n+1)*f[x]*y[x]-f[x],y[x],x,IncludeSingularSolutions -> True]
 

Not solved