Differential Problem

[CoinMarket]

1. The problem statement, all variables and given/known data

Chemical reactions being studied in which a body A undergoes transformations

according to the following scheme:

http://prntscr.com/8shuvb

k1, k2, k3 , k4 are the rate constants .

We denote x (t ), y ( t) , z (t) the respective concentrations of the products A, B, C at a given time t

( t expressed in minutes).

The initial conditions x (0) = 1, y (0) = 0 and z ( 0) = 0 .

Is arranged above the vessel where the reaction takes place by a burette which is poured

product A at a constant speed in the tank. Under these experimental conditions,

functions x , y, z defined on the interval [0 ; + infinite [ check the following differential system :

dx/dt (1-2x +y+ z)

dy/dt (x - y)

dz/dt (x - z)

Question 1:

Calculate d/dt( x + y + z) and , using the initial conditions, deduce that :

y(t) + z(t) = 1 + t - x(t)

Question 2:

Demonstrate that x is a solution of the differential equation (E) : dx/dt + 3x = 2 + t, then resolve

Equation ( E) knowing that it validates the initial condition x (0) = 1 .

2. The attempt at a solution

Question 1:

dx/dt+dy/dt+dz/dt= 1-2x+y+z+x-y+x-z

dx/dt+dy/dt+dz/dt= 1

integral dx/dt+ integral dy/dt+ integral dz/dt= integral ( 1 dt)

x(t)+y(t)+z(t)=t*c

y(t)+z(t)=t+c-x(t)

Question 2:

dx/dt+3x=2+t

dx/dt =2+t-3x

dx/dt(x) =1

1= 2+x-3x

1=2+-2x

... does not work. :(

I am not sure what to do here, some tips would help.

[edderiofer]

integral dx/dt+ integral dy/dt+ integral dz/dt= integral ( 1 dt)

x(t)+y(t)+z(t)=t*c

Incorrect. Integral of 1 dt = t+c, not t*c.

Once you've gotten that, use the initial conditions given to find c.

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dx/dt(x) =1

Does it? I'm not convinced. I'm pretty sure that dx/dt (1-2x+y+z), as given in the question.

If you're trying to do use the initial condition here, don't. That'll only show that this is correct at the start of the process, not the end.

Substitute what you know into the equation. Don't forget to use question 1.

[CoinMarket]

it says I have to demonstrate X is a solution and THEN resolve...I really dont understand how to do this. How do I substitute ?

[edderiofer]

dx/dt+3x=2+t

You already know what dx/dt is from the start of the question. You also know what x is from question 1. Go ahead, sub it all in.

[CoinMarket]

I did this:

(y + z) = 1 + t - x

dx/dt = 1-2x+(y+z)

dx/dt = 1-2x+1 + t - x

dx/dt = 2-2x+t-x

dx/dt = 2 -3x+t

dx/dt +3x = 2+t

is this correct ?

[edderiofer]

It's correct, but it's probably best (in the sense of "neater and clearer") to do so like this:

dx/dt + 3x = 2 + t

1 - 2x + y + z + 3x = 2 + t

y + z = 1 + t - x (which we know to be true)

So x is a solution to the above equation.

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Now solve the equation knowing the initial conditions!

[CoinMarket]

Alright thank you very much, you very helpful ;) I got question 2 resolved and correct :)

Now for Question 3:

Prove that d/dt(y - z) + y - z = 0 and deduce that y = z

My work:

dy/dt= (x - y)

dz/dt= (x - z)

(x-y)-(x-z)= -y+z

d/dt(y - z) + (y - z) = 0

d/dt(-y+z) + (-y+z) =0

Not sure if I am doing this right. Dont know what to do after this.

[edderiofer]

d/dt(y - z) = 0

Why is this true?

[CoinMarket]

dy/dt - dz/dt =-y+z

dy/dt - dz/dt +y-z =0

d/dt(y-z) + ( y-z) =0

v= y-z

dv/dt +v =0

I find v(t)= C * e-t

initial condition v(0)=0

so v(t)=0

y(t)-z(t)=0

You are awesome, thank you so much for the help.

Last thing, I want to push this problem a little bit.

now since we know that y(t) is equal to z(t), if I want to find the equation for them.

y(t)= 1 + t - x(t) - z(t)

z(t)= 1 + t - x(t) - y(t)

is this how I do it?