61.19.8 problem 8

Internal problem ID [12277]
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
Date solved : Tuesday, January 28, 2025 at 07:53:59 PM
CAS classification : [_Riccati]

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

✓ Solution by Maple

Time used: 0.090 (sec). Leaf size: 65

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)
 

\[ y = -\frac {\left (a^{2}+\tanh \left (\frac {\sqrt {a^{2} \left (a^{2}-4 b \right )}\, \left (a \left (\int f x^{n -1}d x \right )+c_{1} \right )}{2 a^{2}}\right ) \sqrt {a^{2} \left (a^{2}-4 b \right )}\right ) x^{-n}}{2 a} \]

✓ Solution by Mathematica

Time used: 1.032 (sec). Leaf size: 82

DSolve[x*D[y[x],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]
 

\[ \text {Solve}\left [\int _1^{\sqrt {\frac {x^{2 n}}{b}} y(x)}\frac {1}{K[1]^2-\sqrt {\frac {a^2}{b}} K[1]+1}dK[1]=\int _1^x\frac {b f(K[2]) \sqrt {\frac {K[2]^{2 n}}{b}}}{K[2]}dK[2]+c_1,y(x)\right ] \]