60.1.65 problem 66

Internal problem ID [10079]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, linear first order
Problem number : 66
Date solved : Wednesday, March 05, 2025 at 08:26:56 AM
CAS classification : [_separable]

\begin{align*} y^{\prime }-\frac {\sqrt {{| y \left (-1+y\right ) \left (-1+a y\right )|}}}{\sqrt {{| x \left (x -1\right ) \left (a x -1\right )|}}}&=0 \end{align*}

✓ Maple. Time used: 0.002 (sec). Leaf size: 40

ode:=diff(y(x),x)-abs(y(x)*(-1+y(x))*(-1+a*y(x)))^(1/2)/abs(x*(x-1)*(a*x-1))^(1/2) = 0; 
dsolve(ode,y(x), singsol=all);
 

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

✓ Mathematica. Time used: 0.454 (sec). Leaf size: 81

ode=D[y[x],x] - Sqrt[Abs[y[x]*(1-y[x])*(1-a*y[x])]]/Sqrt[Abs[x*(1-x)*(1-a*x)]]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {1}{\sqrt {| (1-K[1]) K[1] (1-a K[1])| }}dK[1]\&\right ]\left [\int _1^x\frac {1}{\sqrt {| (K[2]-1) K[2] (a K[2]-1)| }}dK[2]+c_1\right ] \\ y(x)\to 0 \\ y(x)\to 1 \\ y(x)\to \frac {1}{a} \\ \end{align*}

✓ Sympy. Time used: 3.124 (sec). Leaf size: 46

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

\[ \int \limits ^{y{\left (x \right )}} \frac {1}{\left |{\sqrt {y \left (y^{2} a - y a - y + 1\right )}}\right |}\, dy = C_{1} + \int \frac {1}{\left |{\sqrt {x \left (a x^{2} - a x - x + 1\right )}}\right |}\, dx \]