Standard linear first order

Pure mathematical questions of the form dy/dx + P(x)y = Q(x) requiring integrating factor, with initial conditions to find particular solutions. No applied context.

79 questions · Standard +0.5

Sort by: Default | Easiest first | Hardest first
AQA FP3 2007 June Q3
8 marks Standard +0.3
3 By using an integrating factor, find the solution of the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } + ( \tan x ) y = \sec x$$ given that \(y = 3\) when \(x = 0\).
AQA Further Paper 1 2020 June Q10
10 marks Challenging +1.2
10
  1. Find the general solution of the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } + \frac { 2 y } { x } = \frac { x + 3 } { x ( x - 1 ) \left( x ^ { 2 } + 3 \right) } \quad ( x > 1 )$$ 10
  2. Find the particular solution for which \(y = 0\) when \(x = 3\)
    Give your answer in the form \(y = \mathrm { f } ( x )\)
AQA Further Paper 1 2024 June Q14
7 marks Challenging +1.2
14 Solve the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } + y \tanh x = \sinh ^ { 3 } x$$ given that \(y = 3\) when \(x = \ln 2\)
Give your answer in an exact form.
AQA Further Paper 2 2022 June Q9
14 marks Standard +0.8
9
  1. A curve passes through the point (5, 12.3) and satisfies the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \left( x ^ { 2 } - 9 \right) ^ { \frac { 1 } { 2 } } + \frac { 2 x y } { x ^ { 2 } - 9 } \quad x > 3$$ Use Euler's step by step method once, and then the midpoint formula $$y _ { r + 1 } = y _ { r - 1 } + 2 h \mathrm { f } \left( x _ { r } , y _ { r } \right) , \quad x _ { r + 1 } = x _ { r } + h$$ once, each with a step length of 0.1 , to estimate the value of \(y\) when \(x = 5.2\)
    Give your answer to six significant figures.
    9
    1. Find the general solution of the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \left( x ^ { 2 } - 9 \right) ^ { \frac { 1 } { 2 } } + \frac { 2 x y } { x ^ { 2 } - 9 } \quad ( x > 3 )$$ 9
  3. (ii) Given that \(y\) satisfies the differential equation in part (b)(i) and that \(y = 12.3\) when \(x = 5\), find the value of \(y\) when \(x = 5.2\) Give your answer to six significant figures.
    [0pt] [3 marks]
    9
  4. Comment on the accuracy of your answer to part (a).