Challenging +1.2 This is a standard second-order linear ODE with constant coefficients requiring both complementary function (solving auxiliary equation m² - 3m - 4 = 0) and particular integral (trying y = A cos 2x + B sin 2x + Cx + D). While it involves multiple steps and careful algebra, it follows a completely routine procedure taught in Further Maths with no novel problem-solving required. The 9 marks reflect length rather than conceptual difficulty, making it moderately above average but well within standard Further Maths territory.
Find the general solution of the differential equation
$$\frac{\mathrm{d}^2y}{\mathrm{d}x^2} - 3\frac{\mathrm{d}y}{\mathrm{d}x} - 4y = \cos 2x + 5x$$
[9 marks]
PI: y = pcos2− 𝑚𝑚x𝑥𝑥+−qs3i𝑚𝑚 4n𝑥𝑥2−x4+=rx0+⇒s𝑚𝑚 =4,−1
𝑦𝑦=𝐴𝐴𝑒𝑒 +𝐵𝐵𝑒𝑒
y'=−2psin2x+2qcos2x+r
y''=−4pcos2x−4qsin2x
−4pcos2x−4qsin2x+6psin2x−6qcos2x−3r
−4pcos2x−4qsin2x−4rx−4s=cos2x+5x
6p−8q=0
−6q−8p=1
3 2
𝑞𝑞 =− ,𝑝𝑝=−
−4r=55⇒0 r=−52 5
4
−3r−4s=0⇒s=15
16
y= Ae−x +Be4x − 2 cos2x− 3 sin2x−5 x+15
25 50 4 16
Obtains correct
Answer
Marks
Guidance
complementary function
1.1b
A1
Uses correct form of
Answer
Marks
Guidance
particular integral
3.1a
M1
Obtains first and second
derivatives of their
Answer
Marks
Guidance
particular integral
1.1a
M1
Substitutes first and
second derivatives of
Answer
Marks
Guidance
particular integral into DE
3.1a
M1
Forms a pair of
simultaneous equations
by comparing
coefficients of cos2xand
Answer
Marks
Guidance
sin2x and solves them
3.1a
M1
Forms a pair of
simultaneous equations
by comparing coefficient
of xand constant term
Answer
Marks
Guidance
and solves them
3.1a
M1
Obtains correct trig
constants or linear
Answer
Marks
Guidance
constants
1.1b
A1
Completes a reasoned
argument to obtain
completely correct
Answer
Marks
Guidance
solution, including y =
2.1
R1
Total
9
Q
Marking instructions
AO
Question 15:
15 | Uses auxiliary equation | 3.1a | M1 | aux. equation
CF: 2
PI: y = pcos2− 𝑚𝑚x𝑥𝑥+−qs3i𝑚𝑚 4n𝑥𝑥2−x4+=rx0+⇒s𝑚𝑚 =4,−1
𝑦𝑦=𝐴𝐴𝑒𝑒 +𝐵𝐵𝑒𝑒
y'=−2psin2x+2qcos2x+r
y''=−4pcos2x−4qsin2x
−4pcos2x−4qsin2x+6psin2x−6qcos2x−3r
−4pcos2x−4qsin2x−4rx−4s=cos2x+5x
6p−8q=0
−6q−8p=1
3 2
𝑞𝑞 =− ,𝑝𝑝=−
−4r=55⇒0 r=−52 5
4
−3r−4s=0⇒s=15
16
y= Ae−x +Be4x − 2 cos2x− 3 sin2x−5 x+15
25 50 4 16
Obtains correct
complementary function | 1.1b | A1
Uses correct form of
particular integral | 3.1a | M1
Obtains first and second
derivatives of their
particular integral | 1.1a | M1
Substitutes first and
second derivatives of
particular integral into DE | 3.1a | M1
Forms a pair of
simultaneous equations
by comparing
coefficients of cos2xand
sin2x and solves them | 3.1a | M1
Forms a pair of
simultaneous equations
by comparing coefficient
of xand constant term
and solves them | 3.1a | M1
Obtains correct trig
constants or linear
constants | 1.1b | A1
Completes a reasoned
argument to obtain
completely correct
solution, including y = | 2.1 | R1
Total | 9
Q | Marking instructions | AO | Marks | Typical solution
Find the general solution of the differential equation
$$\frac{\mathrm{d}^2y}{\mathrm{d}x^2} - 3\frac{\mathrm{d}y}{\mathrm{d}x} - 4y = \cos 2x + 5x$$
[9 marks]
\hfill \mbox{\textit{AQA Further Paper 1 2023 Q15 [9]}}