# Find constant in exponential

We're given this equation and told to find the value of $A$.

$$k_aAe^{{k_a}t} + k_cBe^{{k_c}t} = k_aCe^{-k_at} + k_c(Ae^{{k_a}t} + Be^{k_ct})$$

Where C is the initial constant of a solution (195). So far I've tried integrating then simplifying (eliminating) both sides of the equation for A, but this is wrong.
$$A=\sqrt{\frac {Ck_a}{k_c}}$$

## 1 Answer

Expanding the brackets on the RHS, the second term cancels out with the second term on the LHS. Then we get $$k_a Ae^{k_at} = k_aCe^{-k_at} + k_cAe^{kt}$$ so moving the right term from the RHS to the LHS, factoring and dividing we get $$\begin{split} k_a Ae^{k_at} - k_cAe^{kt} &= k_aCe^{-k_at} \\ A \left( k_a e^{k_at} - k_ce^{kt}\right) &= k_aCe^{-k_at} \\ A &= \frac{k_aCe^{-k_at}}{k_a e^{k_at} - k_ce^{kt}} \end{split}$$

• Thanks so much! Apr 8, 2018 at 4:53