How to analytically solve for the impulse response function of an ODE system?

Question

I'm quite new to Mathematica, and I'm quite rusty on my differential equations, so I apologise in advance if this is trivial!

I'm trying to use NDSolve to analytically solve a set of differential equations for a compartmental model over time (t) with an input function. Effectively, I want to figure out the impulse-response function of the following model:

 Input  |  System:  (CT = C1 + C2)
        |        
  ____  |  K1   ____    k3   ____   
 | C0 | |----> | C1 | ----> | C2 |  
 |____| |<---- |____| <---- |____|  
        |  k2           k4          
        |

where C0(t) is our known input function, measured over time (however for this exercise, I'm interested in C0(t) as if it were a Dirac delta function). We also measure the total of C1(t)+C2(t) = CT(t) (T for total). Hence, the convolution of our input function C0(t) and the impulse response function, H, generates our measured curve CT(t):

CT(t) = (H * Cp)(t)

The differential equations are as follows:

  dC1(t)/dt = K1*C0(t) - (k2+k3)*C1(t)
  dC2(t)/dt = k3*C1(t) - k4*C2(t)  

I'm not quite sure how to go about solving this in Mathematica. Ideally, I would like to simplify it to a sum of two exponential functions, as was done here (page 648, screenshotted below):

Is there any way I do this? I've been trying the following:

NDSolve[{C1'[t] == K1*C0[t] - (k2 + k3)*C1[t] + k4*C2[t],
         C2'[t] == k3*C1[t] - k4*C2[t],
         C1[0] == 0,
         C2[0] == 0}, {C1, C2}, {t, 0, 120}]

but I get the error:

NDSolve::underdet: There are more dependent variables, {C0[t],C1[t],C2[t]}, than equations, so the system is underdetermined.

I just want to be able to tell Mathematica that C0 is known, and that I only want C1 and C2, which can be convolved with the input function.

Any help would be greatly appreciated!

Answer 1

DSolve will give you a symbolic result, albeit messy and containing integrals containing the arbitrary function C0[t]:

DSolve[{C1'[t] == K1*C0[t] - (k2 + k3)*C1[t] + k4*C2[t],    C2'[t] == k3*C1[t] - k4*C2[t], C1[0] == 0, C2[0] == 0}, {C1[t],    C2[t]}, t]