MYTUTOR SUBJECT ANSWERS

443 views

What are differential equations, and why are they important?

Let's begin by reviewing a fairly simple (linear) equation:

(1) y = 2x + 3

(1) describes the relationship between two variables, x and y. Given a value for one, we can use (1) to work out the value of the unknown. For instance, when x=4, y=11.

Differential equations are more general. They include derivative terms, denoted dy/dx or y'. Let us consider an example:

(2) y' = dy/dx = ky

While (1) described the relation between two variables, (2) describes the relation between y's slope (with respect to x) and y itself. In English, it says that "y's slope is directly proportional to y." Unlike (1), the solution to (2) is a function y, that satisfies the above description. But what kind of function has these properties? Let us solve (2) to find out:

* Take all the 'x' terms to one side, and all the ‘y’ terms to the other side:

dy/dx = ky

dy = ky dx

dy/y = k dx

(1/y) dy = k dx

* Take the integral of both sides (we omit some details about definite integrals and boundary conditions, for simplicity):

INTEGRAL( (1/y) dy) = INTEGRAL(k dx)

* Evaluate the above integral:

ln(y) = kx

* Which is true if and only if:

ekx = y

Thus we have shown that the exponential function y(x) = ekx satisfies the differential equation (2). This is what we should expect, since (2) said "y's slope is directly proportional to y" and this is exactly how the number e is defined.

In further study, you will find that differential equations like (2) have applications in diverse fields such as: in Economics, to model growth rates and continuously compounded interest; Physics, to model radioactive decay or damped oscillations; in Biology to model bacteria populations. Now let us consider a more difficult example:

(3)2ψ/∂x2 = (1/v2)(∂2ψ/∂t2)

(3) looks a little daunting but it is not much more complicated than (2). The solution to (3) needs to be a function ψ(x, t). x stands for a spatial direction, and t is time. v stands for ‘speed.’ We are given details about how ψ’s second (partial) derivatives, with respect to x and t, relate to each other. (3) is known to Physicists as the 'Wave Equation', because it can be derived by carefully studying springs in oscillation and sine/cosine waves like (4) are solutions to it.

(4) ψ = Asin(kx - ωt)

Let us verify this.

* Calculate (4)’s second partial derivative with respect to x (we will need the Chain Rule):

(5)2ψ/∂x2= -k2Asin(kx - ωt)

* Calculate (4)’s second partial derivative with respect to t:

(6)2ψ/∂t2= -ω2Asin(kx - ωt)

* We also need the following result (known as a dispersion relation). In physics, It can be shown that:

(7) ω = kv

where ω is ‘angular frequency’, k is ‘wavenumber’ and v is the speed that a wave propagates at.

* Substituting (5) and (6) into (3) gives:

-k2Asin(kx - ωt) = -(1/v2)ω2Asin(kx - ωt)

-k2 = -(1/v2)ω2

k2/ω2 = 1/v2

v2 = ω2/k2

kv = ω

* Which we know to be correct because of result (7), which follows from the definitions of ‘angular frequency’, ‘wavenumber’ and ‘wave speed’.

Hence the function ψ(x, t) = Asin(kx - ωt) is a solution to the differential equation (3). In fact, Schrödinger’s equation is a famous wave equation, incorporating complex numbers, whose solution is the ‘wavefunction’ of a system of particles. So I hope you can see, from this brief introduction, that we can get a lot of mileage out of this concept of ‘differential equations’!

 

Xavier D. GCSE Economics tutor, A Level Economics tutor, IB Economics...

2 years ago

Answered by Xavier, an A Level Further Mathematics tutor with MyTutor


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

88 SUBJECT SPECIALISTS

£20 /hr

Elena S.

Degree: Mechanical Engineering MEng (Masters) - Imperial College London University

Subjects offered:Further Mathematics , Physics+ 3 more

Further Mathematics
Physics
Maths
Chemistry
Biology

“I am currently studying Mechanical Engineering at Imperial College London. This comes out of my passion for forming real life solutions to problems and love of problem solving. It’s this that inspires my tutoring also. Throughout univ...”

George S. GCSE Further Mathematics  tutor, A Level Further Mathematic...
£20 /hr

George S.

Degree: Physics (Masters) - Birmingham University

Subjects offered:Further Mathematics , Physics+ 2 more

Further Mathematics
Physics
Maths
.PAT.

“About Me: I am just entering my first year to study Physics at University of Birmingham, I've always enjoyed anything maths based hence my A-level choices, I hope I can help you through your work whether it's GCSE or A-level and maybe...”

£20 /hr

Sara D.

Degree: Computer Science and Mathematics (Bachelors) - Edinburgh University

Subjects offered:Further Mathematics , Physics+ 2 more

Further Mathematics
Physics
Maths
Computing

“Top tutor from the renowned Russell university group, ready to help you improve your grades.”

About the author

Xavier D.

Currently unavailable: until 23/05/2016

Degree: Computer Science (Masters) - University College London University

Subjects offered:Further Mathematics , Science+ 6 more

Further Mathematics
Science
Physics
Philosophy and Ethics
Maths
Economics
Chemistry

“My background I study postgraduate Computer Science at University College London, and before that I graduated from the University of Edinburgh reading Philosophy & Economics. I truly believe everybody has the potential to excel in ma...”

MyTutor guarantee

You may also like...

Other A Level Further Mathematics questions

Find the complex number z such that 5iz+3z* +16 = 8i. Give your answer in the form a + bi, where a and b are real numbers.

Find the general solution for the determinant of a 3x3 martix. When does the inverse of this matrix not exist?

What does it mean if two matrices are said to be commutative?

What are polar coordinates?

View A Level Further Mathematics 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