Standard linear first order - variable coefficients

Linear first order ODEs of the form dy/dx + P(x)y = Q(x) where P(x) is a non-constant function of x (e.g., 1/x, tan x, rational functions), requiring computation of a non-trivial integrating factor.

69 questions · Standard +0.7

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Edexcel FP2 Q6
11 marks Standard +0.8
  1. Find the general solution of the differential equation $$\cos x \frac{dy}{dx} + (\sin x)y = \cos^3 x.$$ [6]
  2. Show that, for \(0 \leq x \leq 2\pi\), there are two points on the \(x\)-axis through which all the solution curves for this differential equation pass. [2]
  3. Sketch the graph, for \(0 \leq x \leq 2\pi\), of the particular solution for which \(y = 0\) at \(x = 0\). [3]
Edexcel FP2 Q37
7 marks Standard +0.8
Find the general solution of the differential equation $$\frac{dy}{dx} + 2y \cot 2x = \sin x, \quad 0 < x < \frac{\pi}{2},$$ giving your answer in the form \(y = f(x)\). [7]
Edexcel FP2 Q42
7 marks Standard +0.3
Find the general solution of the differential equation $$(x + 1)\frac{dy}{dx} + 2y = \frac{1}{x}, \quad x > 0.$$ giving your answer in the form \(y = f(x)\). [7]
OCR FP3 Q8
10 marks Standard +0.8
  1. Find the general solution of the differential equation $$\frac{dy}{dx} + y\tan x = \cos^3 x,$$ expressing \(y\) in terms of \(x\) in your answer. [8]
  2. Find the particular solution for which \(y = 2\) when \(x = \pi\). [2]
OCR FP3 Q5
9 marks Standard +0.8
  1. Find the general solution of the differential equation $$\frac{dy}{dx} - \frac{y}{x} = \sin 2x,$$ expressing \(y\) in terms of \(x\) in your answer. [6] In a particular case, it is given that \(y = \frac{2}{\pi}\) when \(x = \frac{1}{4}\pi\).
  2. Find the solution of the differential equation in this case. [2]
  3. Write down a function to which \(y\) approximates when \(x\) is large and positive. [1]
OCR FP3 2008 January Q5
9 marks Standard +0.8
  1. Find the general solution of the differential equation $$\frac{dy}{dx} - \frac{y}{x} = \sin 2x,$$ expressing \(y\) in terms of \(x\) in your answer. [6]
In a particular case, it is given that \(y = \frac{2}{\pi}\) when \(x = \frac{1}{4}\pi\).
  1. Find the solution of the differential equation in this case. [2]
  2. Write down a function to which \(y\) approximates when \(x\) is large and positive. [1]
OCR FP3 2011 January Q1
6 marks Standard +0.3
  1. Find the general solution of the differential equation $$\frac{dy}{dx} + xy = xe^{\frac{x^2}{2}},$$ giving your answer in the form \(y = f(x)\). [4]
  2. Find the particular solution for which \(y = 1\) when \(x = 0\). [2]
OCR FP3 2006 June Q4
8 marks Standard +0.8
Find the solution of the differential equation $$\frac{dy}{dx} - \frac{x^2y}{1 + x^3} = x^2$$ for which \(y = 1\) when \(x = 0\), expressing your answer in the form \(y = f(x)\). [8]
AQA Further Paper 1 2019 June Q11
7 marks Challenging +1.2
Find the general solution of the differential equation $$x \frac{dy}{dx} - 2y = \frac{x^3}{\sqrt{4 - 2x - x^2}}$$ where \(0 < x < \sqrt{5} - 1\) [7 marks]
AQA Further Paper 1 2024 June Q14
7 marks Challenging +1.2
Solve the differential equation $$\frac{dy}{dx} + y\tanh x = \sinh^3 x$$ given that \(y = 3\) when \(x = \ln 2\) Give your answer in an exact form. [7 marks]
AQA Further Paper 1 Specimen Q6
7 marks Standard +0.8
  1. Obtain the general solution of the differential equation $$\tan x \frac{dy}{dx} + y = \sin x \tan x$$ where \(0 < x < \frac{\pi}{2}\) [5 marks]
  2. Hence find the particular solution of this differential equation, given that \(y = \frac{1}{2\sqrt{2}}\) when \(x = \frac{\pi}{4}\) [2 marks]
OCR MEI Further Pure Core Specimen Q10
9 marks Standard +0.8
  1. Obtain the solution to the differential equation $$x \frac{dy}{dx} + 3y = \frac{1}{x}, \text{ where } x > 0,$$ given that \(y = 1\) when \(x = 1\). [7]
  2. Deduce that \(y\) decreases as \(x\) increases. [2]
WJEC Further Unit 4 2019 June Q10
8 marks Challenging +1.8
Given the differential equation $$\sec x \frac{\mathrm{d}y}{\mathrm{d}x} + y\cos \sec x = 2$$ and \(x = \frac{\pi}{2}\) when \(y = 3\), find the value of \(y\) when \(x = \frac{\pi}{4}\). [8]
WJEC Further Unit 4 2023 June Q9
8 marks Standard +0.8
Consider the differential equation $$\left(x+1\right)\frac{\mathrm{d}y}{\mathrm{d}x} + 5y = (x+1)^2, \quad x > -1.$$ Given that \(y = \frac{1}{4}\) when \(x = 1\), find the value of \(y\) when \(x = 0\). [8]
WJEC Further Unit 4 2024 June Q3
9 marks Challenging +1.2
Given the differential equation $$\cos x \frac{\mathrm{d}y}{\mathrm{d}x} + y \sin x = 4 \cos^2 x \sin x + 5$$ and \(y = 3\sqrt{2}\) when \(x = \frac{\pi}{4}\), find an equation for \(y\) in terms of \(x\). [9]
WJEC Further Unit 4 Specimen Q10
11 marks Standard +0.8
Consider the differential equation $$\frac{dy}{dx} + 2y \tan x = \sin x, \quad 0 < x < \frac{\pi}{2}.$$
  1. Find an integrating factor for this differential equation. [4]
  2. Solve the differential equation given that \(y = 0\) when \(x = \frac{\pi}{4}\), giving your answer in the form \(y = f(x)\). [7]
SPS SPS FM Pure 2023 February Q10
10 marks Challenging +1.3
  1. Find the general solution of the differential equation $$\frac{dy}{dx} + \frac{2y}{x} = \frac{x+3}{x(x-1)(x^2+3)} \quad (x > 1)$$ [8]
  2. Find the particular solution for which \(y = 0\) when \(x = 3\). Give your answer in the form \(y = f(x)\). [2]
SPS SPS FM Pure 2024 February Q12
7 marks Challenging +1.2
Find the general solution of the differential equation $$x\frac{dy}{dx} - 2y = \frac{x^3}{\sqrt{4 - 2x - x^2}}$$ where \(0 < x < \sqrt{5} - 1\) [7 marks]
Pre-U Pre-U 9795/1 2018 June Q5
8 marks Standard +0.8
Find, in the form \(y = f(x)\), the solution of the differential equation \(\frac{dy}{dx} + y\tanh x = 2\cosh x\), given that \(y = \frac{3}{4}\) when \(x = \ln 2\). [8]