29.12.7 problem 326

Internal problem ID [4926]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Various 12
Problem number : 326
Date solved : Sunday, March 30, 2025 at 04:15:06 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 }+k \left (x +y-a \right ) \left (x +y-b \right )+y^{2}&=0 \end{align*}

✓ Maple. Time used: 0.006 (sec). Leaf size: 54

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

\[ y = \frac {\left (\left (-x +b \right )^{k +1}+c_1 \left (-x +a \right )^{k} \left (-x +a \right )\right ) k}{\left (k +1\right ) \left (c_1 \left (-x +a \right )^{k}+\left (-x +b \right )^{k}\right )} \]

✓ Mathematica. Time used: 60.329 (sec). Leaf size: 99

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

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

✓ Sympy. Time used: 15.066 (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)
 

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