537 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).

1 year ago

Answered by Edmond, an A Level Maths tutor with MyTutor

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

#### 347 SUBJECT SPECIALISTS

£36 /hr

Degree: Mathematics (Masters) - Bristol University

Subjects offered:Maths, Physics+ 3 more

Maths
Physics
Italian
Further Mathematics
Economics

“Exam Time! I can provide a rapid revision class in any maths module which will test your son or daughter in all the fundamentals”

£36 /hr

Degree: Architecture and Environmental Engineering (Masters) - Nottingham University

Subjects offered:Maths, Physics+ 2 more

Maths
Physics
Design & Technology
-Personal Statements-

“Hi there, I have a passion for helping students achieve, and believe that with my 200+ hours of experience, we will be able to surpass the grades you want!”

£20 /hr

Degree: Biomedical science (Bachelors) - Sussex University

Subjects offered:Maths, Further Mathematics + 4 more

Maths
Further Mathematics
Chemistry
Biology
-Personal Statements-
-Medical School Preparation-

“For the past 5 years I have been tutoring, individuals to small classes. I am passionate about teaching and think it is an essential part of life and careers.”

MyTutor guarantee

|  2 completed tutorials

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

Differentiate y=x/sin(x)

What are stationary points and how do I find them?

A man travels 360m along a straight road. He walks for the first 120m at 1.5ms-1, runs the next 180m at 4.5ms-1, and then walks the final 60m at 1.5ms-1. A women travels the same route, in the same time. At what time does the man overtake the women?

Differentiate the equation y = x^2 + 3x + 1 with respect to x.

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