# How do I solve these coupled Differential Equation?

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1. Sep 29, 2015

### baouba

1. The problem statement, all variables and given/known data
dNa/dt = -Na/Ta where Na is the function and Ta is the constant
dNb/dt = Na/Ta - Nb/Tb where Nb is the function and Tb is the constant

2. Relevant equations
My Prof said Nb(t) has the form Nb(t) = Cexp(-t/Ta) + Dexp(-t/Tb)

3. The attempt at a solution
I know the first equation solves to Na(t) = Na(0)exp(-t/Ta)

The second equation can be written,

dNb/dt = (TbNa-TaNb)/(TaTb)

(TaTb)dNb/dt = TbNa-TaNb

Separation of variables and integrating gives:

(TaTb) ∫ [TbNa-TaNb]^-1 dNb = ∫dt

(TaTb)(-1/Ta)ln(NaTb - NbTa) = t + C

Rearranging,

Nb = (Tb/Ta)Na - Cexp(-t/Tb)

Subbing in Na,

Nb = (Tb/Ta)(Na(0)exp(-t/Ta)) - Cexp(-t/Tb)

at t = 0,

Nb(0) = (Tb/Ta)(Na(0)) - C

so C = (Tb/Ta)(Na(0)) - Nb(0)

Subbing back into Nb(t),

Nb = [(Tb/Ta)(Na(0)exp(-t/Ta))] - [(Tb/Ta)(Na(0)) - Nb(0)]exp(-t/Tb)

Nb = (1 / ((Ta/Tb)-1.)) Na(0)exp(-t/Ta)+ (Nb(0)-(Na(0)/((Ta/Tb)-1.)))exp(-t/Tb);

but I just can't seem to get it. Can anyone tell me where I went wrong?

Thanks

2. Sep 29, 2015

### Ray Vickson

Once you have solved the first equation for $N_a(t)$, your second equation becoms
$$\frac{dN_b}{dt} = f(t) - \tau_b N_b,$$
where $\tau_b \equiv 1 / T_b$ is a constant and $f(t) = N_a(t)/T_a$ is a known function. You can solve this DE using an integrating-factor approach; see, eg.,
https://en.wikipedia.org/wiki/Integrating_factor .