Separable variables - standard (polynomial/exponential x-side)

Questions requiring separation of variables where the x-side integrates using standard polynomial, exponential, or simple trigonometric techniques, with given initial conditions.

55 questions · Standard +0.0

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WJEC Unit 3 2024 June Q14
7 marks Standard +0.3
  1. Given that \(y = \frac{1 + \ln x}{x}\), show that \(\frac{dy}{dx} = \frac{-\ln x}{x^2}\). [2]
  2. Hence, solve the differential equation $$\frac{dx}{dt} = \frac{x^2 t}{\ln x},$$ given that \(t = 3\) when \(x = 1\). Give your answer in the form \(t^2 = g(x)\), where \(g\) is a function of \(x\). [5]
SPS SPS FM Pure 2021 June Q15
7 marks Standard +0.3
The height \(x\) metres, of a column of water in a fountain display satisfies the differential equation \(\frac{dx}{dt} = -\frac{8\sin 2t}{3\sqrt{x}}\), where \(t\) is the time in seconds after the display begins. Solve the differential equation, given that initially the column of water has zero height. Express your answer in the form \(x = f(t)\) [7 marks]
SPS SPS FM Pure 2022 June Q11
8 marks Standard +0.8
Solve the differential equation $$2\cot x \frac{dy}{dx} = (4 - y^2)$$ for which \(y = 0\) at \(x = \frac{\pi}{3}\), giving your answer in the form \(\sec^2 x = g(y)\). [8]
Pre-U Pre-U 9794/2 2012 June Q8
6 marks Moderate -0.3
Solve the differential equation \(\frac{dy}{dx} = -y^2 x^3\), where \(y = 2\) when \(x = 1\), expressing your solution in the form \(y = f(x)\). [6]
Pre-U Pre-U 9794/2 Specimen Q8
14 marks Standard +0.8
    1. Find the general solution of the differential equation $$x \frac{dy}{dx} = y(1 + x \cot x),$$ expressing \(y\) in terms of \(x\). [5]
    2. Find the particular solution given that \(y = 1\) when \(x = \frac{1}{2}\pi\). [2]
  2. The real variables \(x\) and \(y\) are related by \(x^2 - y^2 = 2ax - b\), where \(a\) and \(b\) are real constants.
    1. Show that \(\frac{dy}{dx} = 0\) can only be solved for \(x\) and \(y\) if \(b \geqslant a^2\). [5]
    2. Show that \(y \frac{d^2y}{dx^2} = 1 - \left(\frac{dy}{dx}\right)^2\). [2]