23.1.342 problem 328

Internal problem ID [4949]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Part II. Chapter 1. THE DIFFERENTIAL EQUATION IS OF FIRST ORDER AND OF FIRST DEGREE, page 223
Problem number : 328
Date solved : Tuesday, September 30, 2025 at 09:03:57 AM
CAS classification : [_separable]

\begin{align*} \left (x -a \right ) \left (x -b \right ) y^{\prime }+k \left (y-a \right ) \left (y-b \right )&=0 \end{align*}

✓ Maple. Time used: 0.036 (sec). Leaf size: 113

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

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

✓ Mathematica. Time used: 0.324 (sec). Leaf size: 70

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

\begin{align*} y(x)&\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {1}{(a-K[1]) (b-K[1])}dK[1]\&\right ]\left [\int _1^x\frac {k}{(a-K[2]) (K[2]-b)}dK[2]+c_1\right ]\\ y(x)&\to a\\ y(x)&\to b \end{align*}

✓ Sympy. Time used: 1.128 (sec). Leaf size: 53

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

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