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The differential equation for $y_2(t)$ is:

$\\frac{dy_2}{dt} = k_a y_1 - k_c y_2$

Substituting $y_1(t) = c_0 e^{-k_at}$ into the differential equation for $y_2(t)$ we obtain:

$\\frac{dy_2}{dt} = k_ac_0e^{-k_a t} - k_cy_2$

We can define a test solution of the form:

$y_2(t) = A e^{-k_a t} + B e^{-k_c t}$

where $A$ and $B$ are constants to be determine.

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Substitute the test solution into the right-hand side of the differential equation for $y_2(t)$.

Tips: Use an underscore to represent a subscript. Use '*' to represent multiplication. Use 'exp()' to represent an exponential.

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Differentiate the test solution to obtain $\\frac{dy_2}{dt}$.

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Set the derivative of the test solution equal to the answer from part a so that you have an expression involving $A$, $B$, $e^{-k_a t}$, $e^{-k_c t}$ on both left- and right-hand sides. You should see the term $c_0$ appear exactly once, on the right-hand side.

[[0]] = [[1]]

Factorise the above equation by grouping all terms involving $e^{-k_at}$ together on each side, and all terms involving $e^{-k_ct}$ together on each side.

$e^{-k_a t}\\times$[[0]]$+e^{-k_c t} \\times$[[1]] $ = e^{-k_a t}\\times$ [[2]] $+e^{-k_ct}\\times$ [[3]]

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Consider the terms multiplying by $e^{-k_a t}$ on each side. These terms inside the brackets on each side must be equal in order for the solution to be valid.

Setting the left-hand group with $e^{-k_a t}$ equal to the right-hand group with $e^{-k_a t}$, determine a function for $A$.

A = [[0]]

Using your values of $k_a = ${k_a}, $k_c = ${k_c} and $c_0 = ${c_0}, calculate the value of $A$.

A = [[1]]

We cannot use the same technique to solve for $B$, unfortunately. We will use our initial condition for $y_2(t)$ to obtain $B$ instead.

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Substitute the function you found for $A$ into the test solution and solve for $B$ as a function of the parameters $k_a$, $k_c$, and $c_0$.

As a reminder, the test solution is:

$$y_2(t) = A e^{-k_a t} + B e^{-k_c t}$$.

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Substitute the functions for $A$ and $B$ into the test solution to obtain $y_2(t)$.

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For what value for $t$ is the amount of Theophylline in the bloodstream maximised? Note: Answer as a function of the parameters $k_a$, $k_c$ and $c_0$.

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