Differential equations and tears...
While writing about the fractions, geometric progressions etc. in an earlier post,
I was tempted to include differential equations as well. But I saved it for today. So, here we go...
After finishing her 1st sem exams, a girl wept for 15 minutes... saying that she didn't do well in her exams.
She managed to score 75% in that sem.
After finishing the 2nd sem exams, she wept for 30 minutes and managed to score 80% in that sem.
After finishing the 3rd sem exams, if she weeps for 45 minutes, what would be her score?
Come up with your best guess.
Just by looking at the ordered-pairs (15, 75), (30, 80), (45, ?) we will be tempted to say "85%".
The first ones to come up with this answer are generally considered as "STREET SMART".
marks = f(weeping_time)
But when a mathematician asks these "STREET SMART" guys to explain, how did they comeup with that number....
some of them might not be able to express how they have concluded so.
And, not so surprisingly, those "SMART" guys might get branded (by some of the co-students/teachers) as "LUCKY" fellows.
This page tries to explain the formal mathematical models to answer such questions.
In this case, the question implies that there is a relation between the number of minutes the girl wept and the
marks she scored. The problem is almost solved if we find out that relation.
This kind of relations are generally represented in the form of y = f(x),
which means that if the value of x is changed, the value of y also gets changed.
In this case, if the girl weeps for more time, she gets more marks.
In other words...
There are a zillion possible ways in assuming f(x). We might be tempted to use all fancy stuff like...
y = f(x) = x*0.3 + 70
y = f(x) = x + x2 + x3
y = f(x) = etan(x)
y = f(x) = cosh-1(x + 2x2)
But, the question is... which of the above assumptions fit into our question, and are easy to calculate, easy to verify, and simple to deduct?
Lets get into trigonometric/exponential stuff later. For time being, lets stick to polynomials.
Again, there are a zillion types of polynomials....
y = a.x + b
y = a.x2 + b.x + c
y = a.x3 + b.x2 + c.x + d
There is a golden rule here. The degree of the polynomial is picked, based on the no. of known sample points.
If we have 'n' sample points, we pick the polynomial of degree 'n-1'
(WHY? Thats a story for some other time! )
In this weeping-time-marks-scored problem, the no. of known sample points are two. When the girl wept for 15 minutes,
she scored 75% and when she wept for 30 minutes, she score 80%.
In other words, we have 2 sample points
(x1, y1) = (15, 75) and (x2, y2) = (30, 80)
So, we chose f(x) to be a polynomial of degree 2-1 = 1
In other words... y = a.x + b
Consider that the two sample points (x1, y1) and (x2, y2) lie on the polynomial we have selected.
It means that...
y1 = a.x1 + b ... or ... 75 = a.15 + b
y2 = a.x2 + b ... or ... 80 = a.30 + b
Solving the above two simultaneous equations for a and b, we get...
a = 1/3 and b = 70
So, y = f(x) = 70 + x/3
In other words, marks = 70 + weeping_time / 3
If the girl wept for 45 minutes for not doing well in the exams,
our formula predicts that she is going to score...
marks = 70 + 45/3 = 85%
All izz well!!! But... where are the differential equations?
Hang on!! They are coming up next.
Lets get back to the known sample points (x1, y1) and (x2, y2)
Change in the x values... dx = x2 - x1 = 15
Change in the y values... dy = y2 - y1 = 5
And that implies...
dx = 3dy
That's how the differential equations looks like.
dy/dx = function(x, y)
In this case, the function is a constant function.
dy / dx = 1/3
What it means is... if x value is changed by 1 unit, then the y value is changed by 1/3rd units.
In other words, for every 3 minutes the girl weeps, she will be scoring 1% more
But, 1% more that what? That's where the initial value comes into the picture.
In other words... what if the girl has not wept at all?
How much is she capable of scoring without the aid of her tears?
That is the exact meaning of the initial value.
Generally, the initial value and the differential equation are given as the problem
and the solution is to find out the relation between x and y.
We have already found out the relation between x and y (Where... x = weeping_time and y = marks_scored)
Let's find out the relation once again, this time, using by solving the differential equation...
dy / dx = 1/3
Rearranging the terms, we get...
3.dy = dx
Integrating on the both sides, we get...
(Whoa!!!! I haven't dealt the integration yet. Next time baby!!!)
3.y = x + constant
Now, consider the fact that the girl scored 75% after weeping for 15 minutes.
Here, recall that... y is the score, and x is the weeping time.
So, if we substitute those values in the equation, we get...
3.marks = weeping_time + constant
3.75 = 15 + constant
constant = 210
In other words...
3.marks = weeping_time + 210
marks = 70 + weeping_time / 3
The exact same equation we got by solving the simultaneous equations.
Now, the most important question!!! Are we done here?
Just use 45 for weeping_time and find out the marks (again) and we are done for today.
What if there are other girls (and boys) who wept for different times and scored different scores?
What if we can't conclude the exact relation between the weeping_time and the marks_scored?
Thats where the curve fitting comes into the picture. And thats going to be our next stop.