Find f(x)

Added: 2026-05-12

[00:00:02]Welcome to another video. This is a calculus problem. In fact, it is a differential equation problem and we will be finding the function f [clears throat] ofx. Now just to make things simple, I might switch f ofx to y so I can explain or relate better to students who are used to dydx because um there's just a way you can see the ending from the beginning in this video.

[00:00:32]Okay, let's get into the video.

[00:00:42]>> [music] >> There are several strategies in solving a differential equation. Remember, a differential equation is an equation involving a function and its derivative.

[00:00:57]So, if you form an equation that involves the derivative, whether it's the first or second derivative and the function, you have a differential equation. Okay. Now we'll be able to solve this by you know when I first saw the problem I immediately know I I immediately noticed that if you move everything to the other side and you differentiate something this is what you're going to get. Let let me show you. So if you move this let me write it without the x. So you have fprime.

[00:01:31]If you move this over, you're gonna get minus f * fprime equals zero.

[00:01:38]Okay, but anytime you have f multiplying fprime, you must have done implicit differentiation.

[00:01:46]Okay, so I could tell that I must have differentiated f2 for me to get f * frime because the derivative of f2 is going to be 2 f * frime. So I need to divide this by 2.

[00:02:04]And you differentiate this is just f. So I know that when you take the derivative of this, you're going to get this exactly. When you take the derivative of this, you're going to get this. So the if I want to find the function I just immediately go to this fact. So I'm no longer in the second derivative I'm in the first derivative. But if you don't have sufficient experience with this you may not see this. So I'm going to solve this the traditional way. Okay? Because from here we're almost done.

[00:02:36]Okay? Cuz this is where we're going to land ultimately and then solve it. But let's go back and solve it the traditional way. I'm going to switch from f to y.

[00:02:47]Okay, I'm gonna say let y be equal to f.

[00:02:56]Then y prime is f prime of x.

[00:03:03]Okay, we don't need to say of x because we know y prime equals so let's say f ofx. Okay, let's do it that way. Okay.

[00:03:12]And y-p prime equals f-prime of x. Okay. So with all of this written, we can now rewrite the problem and say that yblep prime is equal to y * y prime.

[00:03:32]I still need to explain some things here because I thought I could just do it.

[00:03:36]All I would do is just integrate both sides.

[00:03:40]Okay, I'm going to explain what's going on here. If I integrate this side and I integrate this side, you're going to say, what am I integrating with respect to? I am still integrating with respect to x, but no x is showing up right now.

[00:03:55]And I'm going to explain what what happens. Okay, so here if you integrate with respect to x, you're going to have y prime as your answer. here. If you integrate with respect to x, you're going to get y^2 / 2 plus a constant. There's a constant here too, but I'm going to move the constant here. Add the two constants together and get this. So, let me call this c prime. Okay.

[00:04:22]Okay. Well, there's no dx. How did you arrive at this answer? Let me explain some things to you.

[00:04:29]You see yp prime is what we call d2y dx 2.

[00:04:41]Okay, which really is ddx of dydx. It means you already got dydx.

[00:04:52]You then differentiated it one more time.

[00:04:56]Okay. So if you integrate a function, if I decide to integrate this, so look, you have yblep prime.

[00:05:06]If I put the integral sign here and I don't put dx, the reason is because I really don't need it to get my y in this answer. But on this side, I'm going to put dx. So you can see what happens to dx. See what happens if I integrate this right hand side and I put dx here. What that dx is going to do is going to help me take out this dx. And what I am left with is the integral of dydx.

[00:05:41]You see this guy and this guy they are inverses of each other. This is integrate the derivative of this. So when you integrate the derivative of a function the two of them cancel each other out.

[00:05:56]>> [snorts] >> So what you have left is dy dx and that's your y prime.

[00:06:05]Okay. So that's the reason why I didn't write all of this. If you differentiate the second one. Now the same thing happens here. Okay. Let's make space for this one. Let me do it here. Okay.

[00:06:17]Notice that the integral of y * y prime is the same thing as the integral of y * dy dx. But since I brought dy dx, I'm going to be integrating integrating with respect to x.

[00:06:35]Look, this dx is going to cancel this dx. So that what I'm actually doing is I'm integrating y with respect to y. If you integrate y with respect to y, what do you get?

[00:06:48]You're going to get y^2 / 2 plus c.

[00:06:54]That's the answer we get here.

[00:06:58]Okay. There are other ways you could do substitution and say let um fp prime of x be another letter and then you start doing chain rule and all that. But this is clean enough for us to get to the answer. Okay, let's get rid of this. So at this point, I need to use the conditions, the initial conditions that we're given. When x equals 0, the function is zero and the derivative of the function when x is zero is 1/2. So let's state that using the initial conditions.

[00:07:44]at x = 0, y prime is 0 and y = 0 also.

[00:07:52]See y oh no y prime is 1/2. Hey Newton, you got to fix that. 1/2 and y is zero.

[00:08:01]So if we use the conditions here, this is going to be zero. This is going to be no, this is going to be 1/2. This would be zero. So we have 12 = 0^ 2 / 2 + c.

[00:08:18]Looking at this, this is going to give us zero. That tells us that c = 1/2. So c is equal to 12.

[00:08:28]Now we can go back to this equation and write things neatly. So therefore we have y prime is equal to y^2 / 2 y^2 over 2 + c 12. So the reason we use the initial condition is to know what c is otherwise oh c prime I called it c prime but now we don't have to do c prime anymore. So now we have this equation as our differential equation.

[00:09:01]So what can we do here? Now this is where I'm going to go back to the normal dydx so you could see what I'm about to do in the end. Okay. So instead of writing y prime, I'm going to write dydx. So we have dydx is equal to I can factor out 1/2. I'm going to end up with 12 of y^ 2 + 1.

[00:09:28]Now we can treat this as a as a separable differential equation. Okay, easy. Okay, now that means we can move this dx here and we can bring this y^2 here. So you want to put all the y parts around dy and anything that doesn't have y, you want to put it on the other side with dx. So I'm going to move y^2 + 1 down here. So I have dy over y^2 + 1 equals on that side I'm going to have half remaining and this dx will go to the other side. Okay. And now I can integrate both sides. If I integrate both sides I'm going to have the integral of dy over y^2 + 1. Okay.

[00:10:17]equals the integral of 12 dx.

[00:10:24]When you integrate this, look the denominator is a square variable 2 + 1 that is arc tan, right? So this is going to be arc tan arc tan of y equals if we integrate any constant, it just introduces the variable. Our variable on this side is x. So it's going to be half of x or we write it as x / 2.

[00:10:52]So if R10 oh there's a constant remember both sides will have constant just move the constant here to this side combine them you get a new constant let's call it C no prime this time no prime C only prime newtons [laughter] okay but this situation now we're rant that we know what the value of C is we go back again to the initial conditions and say when x is zero The derivative is 1/2. The function itself is 0. So here the function is it's going to be zero. So we're going to have arc tan of 0 equ= 0 / 2 + c. But arc tan 0 is 0.

[00:11:40]0 / 2 is 0. So it means c is zero. Okay.

[00:11:44]So we got c is equal to zero.

[00:11:49]So we can go back and write this expression without the C because there's no more C. So we can say arc tan I'm going to write it this way. I know some people don't like this notation but I just want to please everybody. Yeah.

[00:12:03]Okay. So this is going to be arc tan of y will be equal to x / 2 + nothing x / 2. So how do we get y? Well, we take the tangent of both sides cuz tangent will remove arct tan and we take the tangent of this. So we say that y is equal to the tangent of x / 2 and that is the function.

[00:12:40]Never stop learning. Those who stop learning stop living. Bye-bye.