Internal problem ID [10611]
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: 8.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_Riccati]
\[ \boxed {y^{\prime } x -x^{2 n} f \left (x \right ) y^{2}-\left (x^{n} f \left (x \right ) a -n \right ) y=f \left (x \right ) b} \]
✓ Solution by Maple
Time used: 0.047 (sec). Leaf size: 64
dsolve(x*diff(y(x),x)=x^(2*n)*f(x)*y(x)^2+(a*x^n*f(x)-n)*y(x)+b*f(x),y(x), singsol=all)
\[ y \left (x \right ) = -\frac {\left (\tanh \left (\frac {\sqrt {a^{4}-4 a^{2} b}\, \left (a \left (\int \frac {x^{n} f \left (x \right )}{x}d x \right )+c_{1} \right )}{2 a^{2}}\right ) \sqrt {a^{4}-4 a^{2} b}+a^{2}\right ) x^{-n}}{2 a} \]
✓ Solution by Mathematica
Time used: 2.272 (sec). Leaf size: 82
DSolve[x*y'[x]==x^(2*n)*f[x]*y[x]^2+(a*x^n*f[x]-n)*y[x]+b*f[x],y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [\int _1^{\sqrt {\frac {x^{2 n}}{b}} y(x)}\frac {1}{K[1]^2-\sqrt {\frac {a^2}{b}} K[1]+1}dK[1]=\int _1^x\frac {b f(K[2]) \sqrt {\frac {K[2]^{2 n}}{b}}}{K[2]}dK[2]+c_1,y(x)\right ] \]