# Question:differential equations

## Question:differential equations

Maple

Problem Statement: In a catalytic reaction that follows a mechanism known as the Michaelis-Menten kinetics, the reactant (S) combines reversibly with a catalyst (E) to form a complex (ES) with a forward and reverse rate constant of k1 and ki, respectively. The complex then dissociates into product (P) with a reaction rate constant of k2 and the catalyst is regenerated.

```            k1     k2
E + S <---> ES --> E + P
ki
where E  = enzyme (i.e., catalyst)
S  = substrate (i.e., reactant)
ES = enzyme-substrate complex')
P  = product
k1, ki, k2 = elementary reaction rate constants
```

The following ODEs describe how the concentration of each of the four species (E, S, ES, P) changes with time.

```       dE/dt  = -k1*E*S + ki*ES + k2*ES
dS/dt  = -k1*E*S + ki*ES
dES/dt =  k1*E*S - ki*ES - k2*ES
dP/dt  =  k2*ES
```

Integrate the above set of equations with the following rate constants and initial conditions:

```      k1 =  1
ki =  1
k2 = 10
E(0)  = E0  = 0.1
S(0)  = S0  = 1.
ES(0) = ES0 = 0.
P(0)  = P0  = 0.
```

1) How do I solve those four in maple?? (tried searching in help, can't write equations correctly keep getting errors even with implicitdiff every example I try works fine.

2) I need to solve and find S(t)

3) How to express S(t) in matchematical rows in maple

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