29.14.20 problem 401

Internal problem ID [4999]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Various 14
Problem number : 401
Date solved : Tuesday, March 04, 2025 at 07:41:59 PM
CAS classification : [_separable]

\begin{align*} y^{\prime } \sqrt {x \left (1-x \right ) \left (-a x +1\right )}&=\sqrt {y \left (1-y\right ) \left (1-a y\right )} \end{align*}

✓ Maple. Time used: 0.001 (sec). Leaf size: 38

ode:=diff(y(x),x)*(x*(1-x)*(-a*x+1))^(1/2) = (y(x)*(1-y(x))*(1-a*y(x)))^(1/2); 
dsolve(ode,y(x), singsol=all);
 

\[ \int \frac {1}{\sqrt {x \left (x -1\right ) \left (a x -1\right )}}d x -\int _{}^{y \left (x \right )}\frac {1}{\sqrt {\textit {\_a} \left (\textit {\_a} -1\right ) \left (a \textit {\_a} -1\right )}}d \textit {\_a} +c_{1} = 0 \]

✓ Mathematica. Time used: 20.042 (sec). Leaf size: 117

ode=D[y[x],x] Sqrt[x (1-x)(1-a x)]==Sqrt[y[x](1-y[x])(1-a y[x])]; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)\to \text {ns}\left (\frac {1}{2} i \sqrt {a} c_1-\operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {1}{\sqrt {x-1}}\right ),\frac {a-1}{a}\right )|\frac {a-1}{a}\right ){}^2 \left (-1+\text {sn}\left (\frac {1}{2} i \sqrt {a} c_1-\operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {1}{\sqrt {x-1}}\right ),\frac {a-1}{a}\right )|\frac {a-1}{a}\right ){}^2\right ) \\ y(x)\to 0 \\ y(x)\to 1 \\ y(x)\to \frac {1}{a} \\ \end{align*}

✓ Sympy. Time used: 3.666 (sec). Leaf size: 36

from sympy import * 
x = symbols("x") 
a = symbols("a") 
y = Function("y") 
ode = Eq(sqrt(x*(1 - x)*(-a*x + 1))*Derivative(y(x), x) - sqrt((1 - y(x))*(-a*y(x) + 1)*y(x)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ \int \limits ^{y{\left (x \right )}} \frac {1}{\sqrt {y \left (y - 1\right ) \left (y a - 1\right )}}\, dy = C_{1} + \int \frac {1}{\sqrt {x \left (x - 1\right ) \left (a x - 1\right )}}\, dx \]