Forming the Differential Equation of a Family

Transcript

0:00Introduction

Hello everyone! Okay, one small puzzle before we start. Take y equals C x squared. Every value of C gives a different curve: C equals one, C equals two, C equals minus three, and so on, infinitely many curves. Now, can one single equation describe every one of these curves at once, with no C anywhere in it? Sounds impossible, right? But it is possible, and that equation is a differential equation. Today we learn how to build it. Forming the equation of a family comes up often in JEE, and once you see the method, it is really simple. So, let's begin.

0:42One constant, one derivative

So, here is our family on screen: y equals C x squared, and each value of C gives one curve. Here C is an arbitrary constant, which means any fixed number we like. Look at the figure. Each curve is one member of the family. The ones opening upward have C positive, and the ones opening downward have C negative. The two orange curves are just two members, picked out in colour. Okay? Now, we want one equation that every curve here satisfies, with no C in it. So tell me, what can we do to get rid of C? Yes, exactly, we differentiate both sides with respect to x! We get y prime equals two C x. The little dash after y is called prime, and y prime means d y by d x, the slope of the curve. Right? But look, C is still sitting there. So we use the family's own equation as well: it says C equals y over x squared. Put that in, and two C x becomes two y over x. Got it? Now multiply both sides by x, and we get x times y prime equals two y. See, there is no C left! Every single parabola in the figure satisfies this one equation. That is our puzzle solved. One constant, and we needed just one derivative, so this is a first order equation. Clear?

2:09The method: n constants, order n

Now, let's turn this into a rule. If a family has n essential arbitrary constants, meaning independent ones, then its differential equation has order n. The double arrow on screen means gives. And the order, remember, is the order of the highest derivative in the equation. Okay? So, how do we form it? Step one is to differentiate the equation of the family n times. One constant, differentiate once; two constants, differentiate twice; and so on. Right? Now count with me. The family's own equation, plus the n derivatives. How many equations is that? Yes, n plus one equations, exactly! Step two is to eliminate the n constants from these n plus one equations. What is left, with no constant in it, is the differential equation of the family. With me? Let us do a quick check. All circles in the plane: x squared plus y squared, plus two g x, plus two f y, plus c, equals zero. Here g, f and c are the constants. So what is the order of their differential equation? The order is three, because there are three essential constants: g, f and c. No need to differentiate even once to answer this. Just count the independent constants. Clear?

3:37The trap: count only essential constants

Now, a classic trap. Look at y equals a times e to the power, x plus b. Two letters, a and b, so two constants, and a second order equation, right? No, careful, that is exactly the trap! Let's split the power. e to the power, x plus b, is e to the power x, times e to the power b. So y equals a times e to the power b, in brackets, times e to the power x. Okay? And a times e to the power b is just one number. Change a or b any way you like, and you only ever change that one number. So there is only one essential constant. Differentiate once, and y prime equals y. The order is one, not two! So before you count, always check whether some constants combine into one. Got it?

4:31Example 1: eliminate A, B from y = Ae^(2x) + Be^(-2x)

Alright, example one. Form the differential equation of y equals capital A times e to the power two x, plus capital B times e to the power minus two x. Here capital A and capital B are the arbitrary constants. Pause the video and try it first. Done? Okay, let's check. How many constants? Two, capital A and capital B, and they cannot be combined into one. So we expect a second order equation, and we differentiate twice. Right? First derivative: y prime equals two capital A times e to the power two x, minus two capital B times e to the power minus two x. That minus sign comes from the chain rule. Okay? Differentiate again. The two dashes, y double prime, mean the second derivative, d squared y by d x squared. y double prime equals four capital A times e to the power two x, plus four capital B times e to the power minus two x. Now look carefully. Does this remind you of something? Yes, exactly, it is four times y! Take four common, and the bracket is our original y. So the answer is y double prime equals four y, with both constants gone. Two constants, order two, just as the rule said. Clear?

5:58Example 2: circles touching the y-axis at the origin

Now a harder one, example two. Form the differential equation of the circles touching the y axis at the origin. This time no equation is given, so we must first write the family ourselves. Pause the video and try it first. Done? Okay, let's check. Let me ask you this: if a circle touches the y axis at the origin, where must its centre be? Yes, the centre must lie on the x axis. The y axis is the tangent at the origin, so the centre sits on the x axis, at the point a comma zero, and the circle passes through the origin. So x minus a, whole squared, plus y squared, equals a squared. Okay? Expand it, and the a squared cancels on both sides. We get x squared plus y squared equals two times a times x. One constant, a, so we expect a first order equation. Right? Differentiate once: two x, plus two y times y prime, equals two a. Divide by two, and x plus y times y prime equals a. So, is this our answer? No, not yet, because a is still there, and each circle has its own a. So use the family's equation to find a: it is x squared plus y squared, over two x. Put it in, and x plus y times y prime equals x squared plus y squared, over two x. With me? Multiply both sides by two x. Two x squared, plus two x y times y prime, equals x squared plus y squared. Now take two x squared to the right side. On the right, x squared minus two x squared leaves minus x squared. So two x y times y prime equals y squared minus x squared. There it is: no a anywhere, and order one. Every circle touching the y axis at the origin obeys this one equation. Got it?

8:05Quick revision: n constants, order n, merge constants first

So, quick revision. n essential constants give an equation of order n. Differentiate n times, then eliminate all n constants; no constant may remain in your answer. And before you count, combine the constants that merge into one, like a times e to the power, x plus b. Now go and try the practice problems on this page. You will find them easy. All the best!