Internal problem ID [10599]
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.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_Riccati]
\[ \boxed {y^{\prime }+\left (n +1\right ) x^{n} y^{2}-x^{n +1} f \left (x \right ) y=-f \left (x \right )} \]
✓ Solution by Maple
Time used: 0.015 (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)
\[ y \left (x \right ) = \frac {x^{-n -1} \left (x^{n +1} {\mathrm e}^{\int \frac {x^{n +1} f \left (x \right ) x -2 n -2}{x}d x}+\left (\int x^{n} {\mathrm e}^{\int x^{n +1} f \left (x \right )d x +\left (-2 n -2\right ) \left (\int \frac {1}{x}d x \right )}d x \right ) n +\int x^{n} {\mathrm e}^{\int x^{n +1} f \left (x \right )d x +\left (-2 n -2\right ) \left (\int \frac {1}{x}d x \right )}d x -c_{1} \right )}{\left (\int x^{n} {\mathrm e}^{\int x^{n +1} f \left (x \right )d x +\left (-2 n -2\right ) \left (\int \frac {1}{x}d x \right )}d x \right ) n +\int x^{n} {\mathrm e}^{\int x^{n +1} f \left (x \right )d x +\left (-2 n -2\right ) \left (\int \frac {1}{x}d x \right )}d x -c_{1}} \]
✗ Solution by Mathematica
Time used: 0.0 (sec). Leaf size: 0
DSolve[y'[x]==-(n+1)*x^n*y[x]^2+x^(n+1)*f[x]*y[x]-f[x],y[x],x,IncludeSingularSolutions -> True]
Not solved