what happened to the other constant of integration? The standard equation for exponential growth and decay starts and is derived like this: d P d t = k P d P P = k d t ∫ d P P = ∫ k d t ln ⁡ | P | = k t + CI don't understand the left hand side at this point, isn't ∫ 1 x d x = ln ⁡ | x | + C? Where did the constant of integration from the left integral go?

Question

what happened to the other constant of integration? The standard equation for exponential growth and decay starts and is derived like this:                    d        P                    d        t              =  k  P                    d        P            P        =  k  d  t  ∫                    d        P            P        =  ∫  k  d  t      ln    ⁡          |        P          |        =  k  t  +  CI don&#039;t understand the left hand side at this point, isn&#039;t   ∫            1      x        d  x  =  ln  ⁡      |    x      |    +  C? Where did the constant of integration from the left integral go?

Lily Travis · Accepted Answer

When you integrate both sides, each has a constant - you&#039;d get, for constants A,B:  ∫                    d        P            P        =  ∫  k  d  t    ⟹    ln  ⁡      |    P      |    +  A  =  k  t  +  BWell, we can subtract A from both sides and define a constant C=B−A; then  ln  ⁡      |    P      |    +  A  =  k  t  +  B    ⟹    ln  ⁡      |    P      |    =  k  t  +  B  −  A  =  k  t  +  CThis combination of constants is often implicit in solving differential equations - you&#039;ll integrate on two sides and then just combine the constants on whichever side of the equation is more convenient.

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