Internal problem ID [9856]
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]
Solve \begin {gather*} \boxed {y^{\prime }+\left (n +1\right ) x^{n} y^{2}-x^{n +1} f \relax (x ) y+f \relax (x )=0} \end {gather*}
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 186
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)
\[ y \relax (x ) = -\frac {\left ({\mathrm e}^{\int \frac {f \relax (x ) x^{n} x^{2}-2 n -2}{x}d x} x^{n} x -\left (\int \left (-x^{n} n \,{\mathrm e}^{\int x^{n +1} f \relax (x )d x -2 n \left (\int \frac {1}{x}d x \right )-2 \left (\int \frac {1}{x}d x \right )}-x^{n} {\mathrm e}^{\int x^{n +1} f \relax (x )d x -2 n \left (\int \frac {1}{x}d x \right )-2 \left (\int \frac {1}{x}d x \right )}\right )d x \right )-c_{1}\right ) x^{-n}}{x \left (\int \left (-x^{n} n \,{\mathrm e}^{\int x^{n +1} f \relax (x )d x -2 n \left (\int \frac {1}{x}d x \right )-2 \left (\int \frac {1}{x}d x \right )}-x^{n} {\mathrm e}^{\int x^{n +1} f \relax (x )d x -2 n \left (\int \frac {1}{x}d x \right )-2 \left (\int \frac {1}{x}d x \right )}\right )d x +c_{1}\right )} \]
✗ 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