Date | November 2021 | Marks available | 3 | Reference code | 21N.3.AHL.TZ0.2 |
Level | Additional Higher Level | Paper | Paper 3 | Time zone | Time zone 0 |
Command term | Show that | Question number | 2 | Adapted from | N/A |
Question
In this question you will be exploring the strategies required to solve a system of linear differential equations.
Consider the system of linear differential equations of the form:
dxdt=x-y and dydt=ax+y,
where x, y, t∈ℝ+ and a is a parameter.
First consider the case where a=0.
Now consider the case where a=-1.
Now consider the case where a=-4.
From previous cases, we might conjecture that a solution to this differential equation is y=Feλt, λ∈ℝ and F is a constant.
By solving the differential equation dydt=y, show that y=Aet where A is a constant.
Show that dxdt-x=-Aet.
Solve the differential equation in part (a)(ii) to find x as a function of t.
By differentiating dydt=-x+y with respect to t, show that d2ydt2=2dydt.
By substituting Y=dydt, show that Y=Be2t where B is a constant.
Hence find y as a function of t.
Hence show that x=-B2e2t+C, where C is a constant.
Show that d2ydt2-2dydt-3y=0.
Find the two values for λ that satisfy d2ydt2-2dydt-3y=0.
Let the two values found in part (c)(ii) be λ1 and λ2.
Verify that y=Feλ1t+Geλ2t is a solution to the differential equation in (c)(i),where G is a constant.
Markscheme
METHOD 1
dydt=y
∫dyy=∫dt (M1)
ln OR A1A1
Note: Award A1 for and A1 for and .
AG
METHOD 2
rearranging to AND multiplying by integrating factor M1
A1A1
AG
[3 marks]
substituting into differential equation in M1
AG
[1 mark]
integrating factor (IF) is (M1)
(A1)
(A1)
A1
Note: The first constant must be , and the second can be any constant for the final A1 to be awarded. Accept a change of constant applied at the end.
[4 marks]
A1
EITHER
(M1)
A1
OR
(M1)
A1
THEN
AG
[3 marks]
A1
M1
OR A1
AG
[3 marks]
M1
A1
Note: The first constant must be , and the second can be any constant for the final A1 to be awarded. Accept a change of constant applied at the end.
[2 marks]
METHOD 1
substituting and their (iii) into M1(M1)
A1
AG
Note: Follow through from incorrect part (iii) cannot be awarded if it does not lead to the AG.
METHOD 2
M1
A1
M1
AG
[3 marks]
seen anywhere M1
METHOD 1
attempt to eliminate M1
A1
AG
METHOD 2
rewriting LHS in terms of and M1
A1
AG
[3 marks]
(A1)
(M1)
(since ) A1
and are and (either order) A1
[4 marks]
METHOD 1
(A1)(A1)
M1
A1
AG
METHOD 2
(A1)(A1)
M1
A1
AG
[4 marks]