4. -/3 points HHCalcCustom1 11.9.012. (a) Find the equilibrium points for the fo

Question

4. -/3 points HHCalcCustom1 11.9.012. (a) Find the equilibrium points for the following system of equations. dx-24x-6xy dt dt (x, y) (smaller x-value) (x, y) (larger x-value) (b) Explain why x-2, y-5 is not an equilibrium point for this system. Both equations are zero at this point. O The x-value is positive but y-value is not negative. Both equations are not equal to zero at this point. O The y-value is positive but x-value is not negative. Only one of the equations is equal to zero at this point. Additional Materials Section 11.9 Submit AnswerSave ProgresS Practice Another Version

Explanation / Answer

a) We have given dx/dt=24x-6xy and dy/dt=8y-8xy

An Equilibrium point is any point that makes all rates 0 simultaneously.

dx/dt=24x-6xy=0 --(1)

dy/dt=8y-8xy=0 --(2)

from equation (1)

24x-6xy=0

x(24-6y)=0

x=0 or (24-6y)=0 implies -6y=-24 implies y=4

plug x=0 into equation (2)

8y-8*0*(y)=0 implies 8y=0 implies y=0

plug y=4 into equation (2)

8*4-8x*4=0

32-32x=0

-32x=-32 implies x=1

An Equilibrium points are

(x,y)=(0,0) for smaller x-values

(x,y)=(1,4) for larger x-values

b) We have given x=2,y=5 and

plug x=2,y=5 into equation (1) and (2)

dx/dt=24x-6xy=0 --(1)

24*2-6*2*5=0 implies 48-60=-12

dy/dt=8y-8xy=0 --(2)

8*5-8*2*5=0 implies 40-80=-40

x=2,y=5 is not an equilibrium point for this system because both equations are not equal to zero at this point.

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