4.10c Integrating factor: first order equations

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]

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]

Two different colours of paint are being mixed together in a container. The paint is stirred continuously so that each colour is instantly dispersed evenly throughout the container. Initially the container holds a mixture of 10 litres of red paint and 20 litres of blue paint. The colour of the paint mixture is now altered by

• adding red paint to the container at a rate of 2 litres per second
• adding blue paint to the container at a rate of 1 litre per second
• pumping fully mixed paint from the container at a rate of 3 litres per second.

Let \(r\) litres be the amount of red paint in the container at time \(t\) seconds after the colour of the paint mixture starts to be altered.

1. Show that the amount of red paint in the container can be modelled by the differential equation $$\frac{dr}{dt} = 2 - \frac{r}{a}$$ where \(a\) is a positive constant to be determined. [2]
2. By solving the differential equation, determine how long it will take for the mixture of paint in the container to consist of equal amounts of red paint and blue paint, according to the model. Give your answer to the nearest second. [6] It actually takes 9 seconds for the mixture of paint in the container to consist of equal amounts of red paint and blue paint.
3. Use this information to evaluate the model, giving a reason for your answer. [1]

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]

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]

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]

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]

A particular radioactive substance decays over time. A scientist models the amount of substance, \(x\) grams, at time \(t\) hours by the differential equation $$\frac{dx}{dt} + \frac{1}{10}x = e^{-0.1t}\cos t.$$

1. Solve the differential equation to find the general solution for \(x\) in terms of \(t\). [3]

Initially there was \(10\) g of the substance.

1. Find the particular solution of the differential equation. [2]
2. Find to \(6\) significant figures the amount of substance that would be predicted by the model at
   1. \(6\) hours, [1]
   2. \(6.25\) hours. [1]

A particle \(P\) of mass 2 kg can only move along the straight line segment \(OA\), where \(OA\) is on a rough horizontal surface. The particle is initially at rest at \(O\) and the distance \(OA\) is 0.9 m. When the time is \(t\) seconds the displacement of \(P\) from \(O\) is \(x\) m and the velocity of \(P\) is \(v\) ms\(^{-1}\). \(P\) is subject to a force of magnitude \(4e^{-2t}\) N in the direction of \(A\) for any \(t \geqslant 0\). The resistance to the motion of \(P\) is modelled as being proportional to \(v\). At the instant when \(t = \ln 2\), \(v = 0.5\) and the resultant force on \(P\) is 0 N.

1. Show that, according to the model, \(\frac{dv}{dt} + v = 2e^{-2t}\). [3]
2. Find an expression for \(v\) in terms of \(t\) for \(t \geqslant 0\). [5]
3. By considering the behaviour of \(v\) as \(t\) becomes large explain why, according to the model, \(P\)'s speed must reach a maximum value for some \(t > 0\). [2]
4. Determine the maximum speed considered in part (c). [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]

The population density \(P\), in suitable units, of a certain bacterium at time \(t\) hours is to be modelled by a differential equation. Initially, the population density is zero, and its long-term value is 5. The model uses the differential equation $$\frac{dP}{dt} - \frac{P}{t(1 + t^2)} = \frac{te^{-t}}{\sqrt{1 + t^2}}$$ Find \(P\) as a function of \(t\). [You may assume that \(\lim_{t \to \infty} te^{-t} = 0\)]. [11]

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]