61.2.63 problem 63

Internal problem ID [11990]
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 : Wednesday, March 05, 2025 at 03:41:18 PM
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*}

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

ode:=(x-a)*(x-b)*diff(y(x),x)+y(x)^2+k*(y(x)+x-a)*(y(x)+x-b) = 0; 
dsolve(ode,y(x), singsol=all);
 

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

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

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

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

✓ Sympy. Time used: 14.404 (sec). Leaf size: 525

from sympy import * 
x = symbols("x") 
a = symbols("a") 
b = symbols("b") 
k = symbols("k") 
y = Function("y") 
ode = Eq(k*(-a + x + y(x))*(-b + x + y(x)) + (-a + x)*(-b + x)*Derivative(y(x), x) + y(x)**2,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = - \frac {k \left (a e^{\frac {k \left (C_{1} a^{2} k + C_{1} a^{2} + C_{1} b^{2} k + C_{1} b^{2} + a k \log {\left (- a + x \right )} + a \log {\left (- a + x \right )} + b k \log {\left (- b + x \right )} + b \log {\left (- b + x \right )}\right )}{a k + a - b k - b}} - b e^{\frac {k \left (2 C_{1} a b k + 2 C_{1} a b + a k \log {\left (- b + x \right )} + a \log {\left (- b + x \right )} + b k \log {\left (- a + x \right )} + b \log {\left (- a + x \right )}\right )}{a k + a - b k - b}} + x e^{\frac {k \left (2 C_{1} a b k + 2 C_{1} a b + a k \log {\left (- b + x \right )} + a \log {\left (- b + x \right )} + b k \log {\left (- a + x \right )} + b \log {\left (- a + x \right )}\right )}{a k + a - b k - b}} - x e^{\frac {k \left (C_{1} a^{2} k + C_{1} a^{2} + C_{1} b^{2} k + C_{1} b^{2} + a k \log {\left (- a + x \right )} + a \log {\left (- a + x \right )} + b k \log {\left (- b + x \right )} + b \log {\left (- b + x \right )}\right )}{a k + a - b k - b}}\right )}{k e^{\frac {k \left (2 C_{1} a b k + 2 C_{1} a b + a k \log {\left (- b + x \right )} + a \log {\left (- b + x \right )} + b k \log {\left (- a + x \right )} + b \log {\left (- a + x \right )}\right )}{a k + a - b k - b}} - k e^{\frac {k \left (C_{1} a^{2} k + C_{1} a^{2} + C_{1} b^{2} k + C_{1} b^{2} + a k \log {\left (- a + x \right )} + a \log {\left (- a + x \right )} + b k \log {\left (- b + x \right )} + b \log {\left (- b + x \right )}\right )}{a k + a - b k - b}} + e^{\frac {k \left (2 C_{1} a b k + 2 C_{1} a b + a k \log {\left (- b + x \right )} + a \log {\left (- b + x \right )} + b k \log {\left (- a + x \right )} + b \log {\left (- a + x \right )}\right )}{a k + a - b k - b}} - e^{\frac {k \left (C_{1} a^{2} k + C_{1} a^{2} + C_{1} b^{2} k + C_{1} b^{2} + a k \log {\left (- a + x \right )} + a \log {\left (- a + x \right )} + b k \log {\left (- b + x \right )} + b \log {\left (- b + x \right )}\right )}{a k + a - b k - b}}} \]