- The concentration of a drug in the bloodstream of a patient, \(t\) hours after the drug has been administered, where \(t \leqslant 6\), is modelled by the differential equation
$$t ^ { 2 } \frac { \mathrm {~d} ^ { 2 } C } { \mathrm {~d} t ^ { 2 } } - 5 t \frac { \mathrm {~d} C } { \mathrm {~d} t } + 8 C = t ^ { 3 }$$
where \(C\) is measured in micrograms per litre.
- Show that the transformation \(t = \mathrm { e } ^ { x }\) transforms equation (I) into the equation
$$\frac { \mathrm { d } ^ { 2 } C } { \mathrm {~d} x ^ { 2 } } - 6 \frac { \mathrm {~d} C } { \mathrm {~d} x } + 8 C = \mathrm { e } ^ { 3 x }$$
- Hence find the general solution for the concentration \(C\) at time \(t\) hours.
Given that when \(t = 6 , C = 0\) and \(\frac { \mathrm { d } C } { \mathrm {~d} t } = - 36\)
- find the maximum concentration of the drug in the bloodstream of the patient.