Page 38 - Pra U STPM 2022 Penggal 2 - Mathematics

Mathematics Semester 2 STPM Chapter 4 Differential Equations

When t = 0, let x = A
⇒ C = ln A
\ ln x = –kt + ln A

ln x = –kt
A
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x = e –kt
A
\ The solution of the differential equation is x = Ae .
–kt

(b) When t = 0, A = D ⇒ x = De –kt
When t = 1, x = 3 D ⇒ 3 D = De –k
4 4
e = 3
–k
4
–kt
–k t
The solution of the differential equation x = De can be written as x = D(e ) .
3
\ x = D 1 2 t
4
(c) After 2 hours:
The first dose of drug administered at time t = 0 will reduce to the amount of
3
D 1 2 2 .
4
The second dose of drug administered at time t = 1 will reduce to the amount
3
of D 1 2 1 .
4 4
The third dose of drug administered at time t = 2 has amount D.
\ Immediately after 2 hours the total amount of drug remaining is
3
5
3
D + D 1 2 + D 1 2 2 = 2 16 D
4
4
(d) Let the amount of drug remaining after an indefinite period be S.
3
3
3
3
S = D + D 1 2 + D 1 2 2 + D 1 2 3 + D 1 2 4 + …
4
4
4
4
3
3
1
2
S = D 1 + 3 + 1 2 1 2 3 + … 2
+
4
4
4
1
S = D 1 1 – 2
3
4
\ S = 4D
146

04 STPM Math(T) T2.indd 146 28/01/2022 5:44 PM