Order and Degree

Transcript

0:00Introduction

Hello everyone! Okay, one small puzzle before we start. Take two equations. In the first, the second derivative of y equals sine of x, okay? In the second, the second derivative of y equals sine of the first derivative. They look like twins, right? But the first one has degree one, and the second one has no degree at all! How can that be? Today we learn the order and the degree of a differential equation, and by the end of this video you will see exactly why. This is the very first step of differential equations, and questions on it come up often in JEE exams. So, let's begin.

0:39Differential equations and their order

So, first, what is a differential equation? It is an equation that involves an unknown function, y of x, and one or more of its derivatives. That's all, okay? Now the notation. y prime, y with one small dash, is d y by d x, the first derivative. y double prime, with two dashes, is d squared y by d x squared, the second derivative. And y triple prime, with three dashes, is the third derivative, d cubed y by d x cubed. We will use both ways of writing. Clear? Every differential equation gets two labels: its order and its degree. The order first. If an equation has both y prime and y double prime in it, which one do you think decides the order? The higher one, yes, exactly! The order is the order of the highest derivative present in the equation. Got it? So, our twins from the puzzle: y double prime equals sine x, and y double prime equals sine of y prime. In both of them, the highest derivative is y double prime, so both have order two. The order cannot tell them apart. For that, we need the degree. With me?

1:55Degree: the power of the highest derivative

Now the degree, and here there is a condition. The equation must first be written as a polynomial in all the derivatives. That means the derivatives appear only with whole number powers, like squared or cubed. Okay? Then the degree is the power of the highest order derivative, in that polynomial form. Right? Let's try one from the course. d squared y by d x squared, in a bracket raised to the power three, plus d y by d x to the power four, plus y, equals zero. Four is the biggest power on the screen. So is the degree four? Or is it three? The degree is three, and if you said three, very good! The four belongs to d y by d x, the first derivative, which is not the highest one. The highest derivative is y double prime, and its power is three. So the order is two, and the degree is three.

2:49How to find the degree: three steps

Clear? So, how do we find the degree of a harder equation? The course gives three steps. Step one: remove the radicals, that is, square roots and cube roots, and any fractional powers of the derivatives, by raising both sides to a suitable power. Okay? Step two: if a derivative sits in a denominator, clear it by multiplying both sides by that derivative, right? And step three: only now, read off the power of the highest order derivative. With me? Now look at sine x, and e to the power y, on the screen. These are functions of x and y alone, with no derivative inside them, so they do not affect the degree at all. Clear? And one more point: the order and the degree, when the degree exists, are always positive integers. Keep this in mind, because it is about to save us from a trap. Okay?

3:45The trap, and equations with no degree

Now the most common trap. The tick-shaped sign you see is the square root sign. So here, the square root of y double prime equals y prime. The highest derivative is y double prime, and under the square root its power is one half. So the degree is one half, right? No, no, be careful here, because we just said the degree is a positive integer. Clear the radical first: square both sides, and we get y double prime equals y prime squared. Now y double prime has power one, so the degree is one. Always clear the radical, then read the degree. Got it? Now back to our twins. y double prime equals sine x has degree one, because sine x does not affect the degree. But the second twin, y double prime equals sine of y prime? Can you square it, or cube it, or do anything at all to make it a polynomial in y prime? No, there is no way to do it! Sine of y prime can never be written as a polynomial in y prime, so its degree is not defined. The same is true for e to the power y double prime. And that solves our puzzle: the same order two, but one twin has degree one, and the other has no degree at all. Right?

4:58Example 1: root of 1 plus y' squared equals k y''

Now let's try one from the course. Find the order and degree of this: the square root of, one plus d y by d x squared, equals k times d squared y by d x squared. Pause the video and try it first. Done? Okay, let's check. The highest derivative here is d squared y by d x squared, the second derivative, so the order is two. Easy, right? Now the degree. d squared y by d x squared appears just once, to the power one. So the degree is one, isn't it? No, no, wait, look at the left side, because it is a square root, and the first derivative sits inside it. Clear it first by squaring both sides: one plus y prime squared equals k squared times y double prime squared. Clear? Now y double prime has power two. So the order is two, and the degree is two. See how the square root was hiding the real degree? That is exactly the trap we just saw.

6:01Example 2: y = x y' + root of 1 plus y' squared

Now a slightly harder one, also from the course. y equals x times d y by d x, plus the square root of, one plus d y by d x squared. Find the order and the degree. Pause the video and try it first. Done? Okay, let's check. The only derivative in this equation is d y by d x, so the order is one. Right? For the degree, the radical must go. But first, move the x times y prime term to the left side, so that the square root stands alone. y minus x times y prime equals the square root of, one plus y prime squared. Okay? Now square both sides: y minus x times y prime, the whole squared, equals one plus y prime squared. So tell me, what is the highest power of y prime now? It is two, yes, exactly! Squaring the bracket gives a y prime squared term, and the right side has y prime squared too. So the equation is quadratic in y prime: the order is one, and the degree is two. And if you read degree one from the original equation, that was the trap, right?

7:16Quick revision: order, degree, and when degree is undefined

So, quick revision. The order is the highest derivative present. The degree is its power, but only after clearing the radicals and fractions of the derivatives. And if the equation cannot be made a polynomial in the derivatives, like sine of y prime, the degree is not defined. Now go and try the practice problems on this page. You will find them easy. All the best!