How can I write the differential equation below?

I want to code the mathematical model below:

could you please help me out?

I want to estimate Q (flow) in this equation.

I also have omega=[7200,8800]

and can assign value to H.

After writing the code, I want to proceed with a simulink model.

Thank you for your help in advance

Answer 1

[Torsten]

With

c1 = (b*omega + c*H)/L
c2 = (a + d*omega^2)/L

the solution Q(t) is

Q(t) = (-c1 + (c1+c2*Q0) * exp((t-t0)*c2)) / c2

where the initial condition is Q(t=t0) = Q0

[seyyed Erfan ghoreyshipour]

Thank you so much dear Torsten.

It worked!

could you please ellaborate a little bit on your answer?

how did you do this?

[Torsten]

By hand.

But you can use dsolve instead:

syms Q(t) t0 Q0 a b c d omega H L   
eqn = L*diff(Q,t) == a*Q + b*omega + c*H + d*omega^2*Q; 
cond = Q(t0) == Q0;
ySol(t) = dsolve(eqn,cond)

[seyyed Erfan ghoreyshipour]

Thank you!

It's great!

One question!

shouldn't it be c2 = ((a + d*omega^2)*Q)/L in the first equation?

[Torsten]

No.

With

c1 = (b*omega + c*H)/L
c2 = (a + d*omega^2)/L

the differential equation reads

dQ/dt = c1 + c2*Q

thus

dQ/(c1+c2*Q) = dt

thus

integral_{Q0}^{Q} dQ/(c1+c2*Q) = integral_{t0}^{t} dt

thus

1/c2 * log((c1+c2*Q)/(c1+c2*Q0)) = t-t0

thus

exp(c2*(t-t0)) = (c1+c2*Q)/(c1+c2*Q0)

I leave it to you to solve for Q now.

Answer 2

[seyyed Erfan ghoreyshipour]

Thank you master!