At t seconds, the temp. of the water is θ°C. The rate of increase of the temp. of the water at any time t is modelled by the D.E. dθ/dt=λ(120-θ), θ<=100 where λ is a pos. const. Given θ=20 at t=0, solve this D.E. to show that θ=120-100e^(-λt)

When solving any differential equation, the first method to consider is the seperation of variables. This is the simplest method and, conveniently, it works in this case. To seperate variables:1. Put all of the same type of variables on their own side. In this case, our two variables are t and θ. So in this case, divide everything by (120-θ) and multiply everything by dt. This leaves us with dθ/(120-θ)=λdt.2. The next step is to integrate both sides.       On the left side, we have a variable ^-1 so our answer must have a natural log of the denominator. So lets work backwards, let's assume our answer is ln(120-θ). If we differentiate this, this gives us -1/(120-θ), which is what we started with, but multiplied by -1. Therefore, we must have -ln(120-θ).       On the right side, this is simply integrating a constant (as λ is a constant), so we have λt +c. REMEMBER TO ADD OUR CONSTANT OF INTEGRATION!       This leaves us with: -ln(120-θ)=λt +c.3. Next we find what c is. This is found by applying the initial conditions given in the question, i.e. θ=20, t=0. Plugging this in and rearranging for c, we have c=-ln(100), leaving us with: -ln(120-θ)=λt-ln(100).4. We now have our final expression, but it isn't in the correct form. Therefore, we must use the rules of exponentials to manipulate it. This will leave us with the correct form of: θ=120-100e^(-λt).

EW
Answered by Edmond W. Maths tutor

8516 Views

See similar Maths A Level tutors

Related Maths A Level answers

All answers ▸

A triangle has sides A, B and C. The side BC has length 20cm, the angle ABC is 50 deg and angle BAC is 68 deg. a) Show that the length of AC is 16.5cm, correct to three significant figures. b) The midpoint of BC is M, hence find the length of AM


Solve the equation 5^x = 8, giving your answer to 3 significant figures.


Solve the following equation: 5x - 1 = 3x + 7


The line L has equation 7x - 2y + 11 = 0, Find the gradient of l


We're here to help

contact us iconContact ustelephone icon+44 (0) 203 773 6020
Facebook logoInstagram logoLinkedIn logo

MyTutor is part of the IXL family of brands:

© 2025 by IXL Learning