MYTUTOR SUBJECT ANSWERS

351 views

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).

Edmond W. GCSE Maths tutor, A Level Maths tutor

9 months ago

Answered by Edmond, an A Level Maths tutor with MyTutor


Still stuck? Get one-to-one help from a personally interviewed subject specialist

294 SUBJECT SPECIALISTS

£26 /hr

Samuel C.

Degree: Physics (Bachelors) - Durham University

Subjects offered:Maths, Physics+ 2 more

Maths
Physics
Further Mathematics
Chemistry

“Hi, I'm Sam Crawford. I'm studying Maths and Physics at Durham University, with an offer from Cambridge for next year, and I absolutely love both subjects.”

£26 /hr

Lloyd S.

Degree: Mathematics G100 (Bachelors) - Bristol University

Subjects offered:Maths, Further Mathematics

Maths
Further Mathematics

“About Me:I am 19 and from Devon, currently in my first year studying Maths at the University of Bristol. I have a real passion for Maths and I really hope I can help you to understand, and maybe even enjoy doing maths - I know not ev...”

£20 /hr

Elizabeth P.

Degree: Geophysics (Bachelors) - Durham University

Subjects offered:Maths, -Personal Statements-

Maths
-Personal Statements-

“Hey there! My name is Libby and I’m currently in my final year studying Geophysics at Durham university. I love to get other people as excited about maths and science as I am and have done so by tutoring at school, being involved outr...”

MyTutor guarantee

About the author

Edmond W. GCSE Maths tutor, A Level Maths tutor

Edmond W.

Currently unavailable: for new students

Degree: Mathematics (Bachelors) - Birmingham University

Subjects offered:Maths

Maths

“Hi! My name is Ed Wong and I'm a third year Mathematics student at the University of Birmingham. I have a great passion for maths, and would love to tutor anyone who is finding it difficult. I would say I am great at explaining my sol...”

You may also like...

Other A Level Maths questions

Solve the equation tanx/cosx = 1 for 0°<x<360°

How do I integrate ln(x)

What is the integral of (cos(x))^2?

Find the equation of the tangent to the curve y = 2 ln(2e - x) at the point on the curve where x = e.

View A Level Maths tutors

We use cookies to improve your site experience. By continuing to use this website, we'll assume that you're OK with this. Dismiss

mtw:mercury1:status:ok