Do you guys know any good website to learn DFQ? Let S(t) represent the amount of

Question

Do you guys know any good website to learn DFQ?

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 and S(0) = S0, where a, K and S0 are positive constants. Obtain an implicit solution of the initial value problem. (The differential equation, often referred to as the Michaelis-Menten equation, arises in the study of biochemical reactions.) Enter the constant S0 as S0 (with a zero, not the letter O) Do you guys know any good website to learn DFQ?

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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