Solve the homogenous ODE below and you arrive at your answer.

** d[P]/dt = k2.N - (k1 + k2)[P] **

Is anyone not familiar with calculus?

This is a first order homogeneous ODE, so we can use the Integrating Factor (IF) technique.

d[P]/dt = k2.N - (k1 + k2)[P]

d[P]/dt + (k1 + k2)[P] = k2.N

IF = exp(int (k1+k2) dx) = exp( (k1+k2)t )

int [[ exp( (k1+k2)t ).d[P]/dt + exp( (k1+k2)t ).(k1 + k2)[P] ]] dt= int [[ exp( (k1+k2)t ).k2.N ]] dt

exp( (k1+k2)t ).[P] = k2.N.exp( (k1+k2)t )/(k1+k2) + C

where C is the constant of integration

**when t = 0, [P] = P**_{0}exp(0).P

_{0} = k2.N.exp(0)/(k1+k2) + C

P

_{0} = k2.N/(k1+k2) + C

C = P

_{0} - k2.N/(k1+k2)

Solving the above equation,

P = [[ (k2.N.exp((k1+k2)t)+P

_{0}.(k1+k2) - k2.N)]] /[[ (k1+k2).exp((k1+k2).t) ]]

**note: exp(x) = e**^{x}