61.2.63 problem 63

Internal problem ID [12069]
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
Date solved : Tuesday, January 28, 2025 at 12:13:25 AM
CAS classification : [_rational, [_1st_order, `_with_symmetry_[F(x),G(x)]`], _Riccati]

\begin{align*} \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{align*}

✓ Solution by Maple

Time used: 0.003 (sec). Leaf size: 54

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)
 

\[ y = \frac {k \left (\left (a -x \right )^{k +1}+c_{1} \left (-x +b \right )^{k} \left (-x +b \right )\right )}{\left (k +1\right ) \left (c_{1} \left (-x +b \right )^{k}+\left (a -x \right )^{k}\right )} \]

✓ Solution by Mathematica

Time used: 2.455 (sec). Leaf size: 101

DSolve[(x-a)*(x-b)*D[y[x],x]+y[x]^2+k*(y[x]+x-a)*(y[x]+x-b)==0,y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to \frac {1}{2} \left (\frac {k (a+b-2 x)}{k+1}+\sqrt {-\frac {k^2 (a-b)^2}{(k+1)^2}} \tan \left (\frac {1}{2} \sqrt {-\frac {k^2 (a-b)^2}{(k+1)^2}} \int _1^x\frac {k+1}{(a-K[5]) (K[5]-b)}dK[5]+c_1\right )\right ) \]