Internal problem ID [9650]
Book: Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev. Second
edition
Section: Chapter 1, section 1.2. Riccati Equation. 1.2.2. Equations Containing Power Functions
Problem number: 63.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_rational, [_1st_order, _with_symmetry_[F(x),G(x)]], _Riccati]
Solve \begin {gather*} \boxed {\left (x -a \right ) \left (x -b \right ) y^{\prime }+y^{2}+k \left (y+x -a \right ) \left (y+x -b \right )=0} \end {gather*}
✓ Solution by Maple
Time used: 0.003 (sec). Leaf size: 128
dsolve((x-a)*(x-b)*diff(y(x),x)+y(x)^2+k*(y(x)+x-a)*(y(x)+x-b)=0,y(x), singsol=all)
\[ y \relax (x ) = \frac {k \left (\frac {\left (-x +b \right )^{k} c_{1} b}{c_{1} \left (-x +b \right )^{k}+\left (-x +a \right )^{k}}-\frac {\left (-x +b \right )^{k} c_{1} x}{c_{1} \left (-x +b \right )^{k}+\left (-x +a \right )^{k}}+\frac {\left (-x +a \right )^{k} a}{c_{1} \left (-x +b \right )^{k}+\left (-x +a \right )^{k}}-\frac {\left (-x +a \right )^{k} x}{c_{1} \left (-x +b \right )^{k}+\left (-x +a \right )^{k}}\right )}{k +1} \]
✓ Solution by Mathematica
Time used: 108.564 (sec). Leaf size: 99
DSolve[(x-a)*(x-b)*y'[x]+y[x]^2+k*(y[x]+x-a)*(y[x]+x-b)==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {1}{2} \left (\frac {k (a+b-2 x)}{k+1}+\sqrt {-\frac {k^2 (a-b)^2}{(k+1)^2}} \tan \left (\frac {(k+1) \sqrt {-\frac {k^2 (a-b)^2}{(k+1)^2}} (\log (x-b)-\log (x-a))}{2 (a-b)}+c_1\right )\right ) \\ \end{align*}