Equation 1: (d[Y])/dt = k1 - v2[Y]*((ϵ^2+([X]/Kn )^2)/(1+([X]/Kn )^2 ))

Equation 2: (d[X])/dt = v2[Y]*((ϵ^2+([X]/Kn )^2)/(1+([X]/Kn )^2 )) - k3[X]

Kn=3 and ϵ=0

How do I find the value of [X] when system is at steady state?

To find the value of [X] when the system is at steady state, we need to set both equations to 0 and solve them simultaneously. At steady state, the rates of change of [X] and [Y] are both equal to zero.

Let's set up the equations:

Equation 1: (d[Y])/dt = k1 - v2[Y]*((ϵ^2+([X]/Kn )^2)/(1+([X]/Kn )^2 ))

Equation 2: (d[X])/dt = v2[Y]*((ϵ^2+([X]/Kn )^2)/(1+([X]/Kn )^2 )) - k3[X]

Now, substitute Kn=3 and ϵ=0 into both equations to simplify them:

Equation 1: (d[Y])/dt = k1 - v2[Y]*((0^2+([X]/3 )^2)/(1+([X]/3 )^2 ))

Equation 2: (d[X])/dt = v2[Y]*((0^2+([X]/3 )^2)/(1+([X]/3 )^2 )) - k3[X]

Next, set both equations to 0:

k1 - v2[Y]*((0^2+([X]/3 )^2)/(1+([X]/3 )^2 )) = 0

v2[Y]*((0^2+([X]/3 )^2)/(1+([X]/3 )^2 )) - k3[X] = 0

Now, solve these equations simultaneously to find the value of [X] at steady state. Since the equations involve complex algebraic terms, it may be challenging to find an analytical solution. In such cases, numerical methods or software tools can be employed to solve this system of equations and find the value of [X] at steady state.