Solve ODE with initial conditions

Question

Solve:

 

dA/dt = -10A

A(0) = 5

 

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I tried integrating both sides:

dA/-10A = dt

int(dA/-10A) = int(dt)

-1/10*int(A^-1 dA) = t + c

 

but I'm not sure what else to do

Answer 1

You made this problem too hard for yourself!  Separate the variables as you did, but leave the coefficients.

Notice that I did not fiddle around with the exact (and irrelevant) relationship between Constant and Konstant!

 

dA/dt = - 10A

 

dA/A = -10 dt

 

ln A = -10t + Constant

 

A = Konstant * e^-10T

 

5 = Konstant * e⁰  => Konstant = 5

 

A= 5e^-10t

Answer 2

dA/dt = -10A    So, (-1/10)(1/A)dA = dt

 

Integrate both sides to get (-1/10)lnA = t + C

 

Since A = 5 when t = 0, (-1/10)ln5 = 0 + C

 

                                                C = (-1/10)ln5

 

So, (-1/10)lnA = t + (-1/10)ln5

 

                lnA = -10t + ln5

 

                  A(t) = e^{-10t + ln5}

 

                  A(t) = e^ln5e^-10t

 

                  A(t) = 5e^-10t