6. Consider the ODE dy/dt = y(y-1)(y-3) Find the equilibrium points of this auto

Question

6. Consider the ODE
dy/dt = y(y-1)(y-3)
Find the equilibrium points of this autonomous ODE and create a phase line to classify
their stability. Sketch a few solution curves for various initial conditions. Then solve the
ODE with initial condition y(0) = 2

An implicit solution is fine!

please be detailed because I have no idea how to do it... thanks!

Explanation / Answer

dy/dt =y(y-1)(y-3) =f(y) the equilibrium points are obtained by f(y*)=0 => y*(y*-1)(y*-3) =0 => y* has the soultions 0,1,3 if d/dy(f(y))0 then the ODE is unstable d/dy(f(y)) =d/dy(y(y-1)(y-3)) =(y)(y-1) +(y-1)(y-3)+ (y)(y-3) = > 3(y^2) -8y +3 the function 3(y^2) -8y +3 is lesser than 0 in the interval ((4-sqrt(7))/3,(4+sqrt(7))/3 ) and is greater than 0 in the remaining therefore the function is stable in the interval ((4-sqrt(7))/3,(4+sqrt(7))/3 ) solving the ODE dy/dt =y(y-1)(y-3) => dy/((y)(y-1)(y-3)) =dt => dy((1/3y) -(1/2(y-1))+ (1/6(y-3))) =dt integrating on both sides gives 1/3(ln(y) )-1/2(ln(y-1)) +1/6(ln(y-3))= t substituing the limits from 0 to t ln((y^1/3)((y-3)^1/6))/(y-1^1/2)2^1/3) =t the solution of the ODE dy/dt =y(y-1)(y-3)

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