Let S(t) represent the amount of a chemical reactant present at time t, where t?

Question

Let S(t) represent the amount of a chemical reactant present at time t, where t?0. Assume that S(t) can be determined by solving the initial value problem

S?=?(aS)/K+S, S(0)=S0,

where a,K, and S0 are positive constants. Obtain an implicit solution of the initial value problem.


I've tried integrating a couple of different ways but I wasn't able to find a way to to get "S" where one wasn't in log form and the other was not. I would appreciate any help. I set up the integration to be...

K*ln(S) + S = - at + c

Explanation / Answer

Step 1: Multiply both sides by inverse of RHS, obtaining:
-[(K+S)/(aS)]dS = dt

Step 2: Factor out constants, separate fraction, distribute negative sign
-K/aS - 1/a dS = dt

Step 3: Integrate
(-K/a)ln(S) - S/a = t + C

Step 4: Multiply both sides by a
-K*ln(S) - S = at + aC

Step 5: Solve for S
S = -at - aC - K*ln(S)

Step 6: Solve for C in terms of S0 (with t=0)
S0 = -aC - K*ln(S0)
S0 + K*ln(S0) = -aC
-(S0 + K*ln(S0))/a = C

Step 7: Substitute C back into [5]
S = -at + S0 + K*(ln(S0)) - K*(ln(S)

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