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Solve the differential equation.$$\frac{1}{x} \frac{d y}{d x}=y e^{x^{2}}+2 \sqrt{y} e^{x^{2}}$$

$$4 \ln (\sqrt{y}+2)=e^{x^{2}}+C$$

Calculus 1 / AB

Calculus 2 / BC

Chapter 7

Integrals and Transcendental Functions

Section 2

Exponential Change and Separable Differential Equations

Functions

Trig Integrals

Missouri State University

Harvey Mudd College

Idaho State University

Lectures

01:11

In mathematics, integratio…

06:55

In grammar, determiners ar…

11:49

Solve the given differenti…

00:45

Solve the differential equ…

07:09

03:18

06:41

01:26

08:23

02:27

02:08

03:45

I want to solve this differential equation here. Now the first thing I'm going to do is looks, Dr all this e to the X squared on the right hand side and something else I'm gonna do that may not seem too intuitive right now is factor all the square root of why, But once we divide everything over, it'll become a little bit more parent as to why I want to do this. So I would have square root of why. Plus two. No, let's go ahead and do all of our cross multiplication stuff. So on the left hand side, we should have one over the square root of Why times the square root of why plus two. Then we have a d y. And on the right hand side that should give us X e to the X squared T X. Now, let's go ahead and integrate each side here. Now, the reason why I wanted to write it like that because now you might notice they use of that we can do that Looks a lot like natural log. So, um, over here, all right. This cause I'll also need to do it. Use up or X E to the X squared pressure. So we have you is equal to the square root of why plus two, which gives us d'you is equal to 1/2 square feet of Why what are one over swearing? Why now? Let's multiply each side by two. So that concert of that 1/2 right there and now we have d Y times whenever it's crude. Why? So that means the right hand side can give us what we want to put an equal sign there. So would be the integral of two. Do you over you, which we know integrates two to natural law of the absolute value of you. But you is the square root of why plus two, Let me write that. Why a little bit better? All right, so now let's just put some equals going down and try to figure out what we need to do about this second interval here. Well, we could do a U sub or actual call it, um be so the is going to be equal to, in this case X squared. So then D B is going to be two x d x, and at this point we would want to divide each side by tubes of those cancel out, and then this is going to become Devi over to. So I'm gonna factor that 1/2 out front and we're going to just have e to the B d b, which just integrates too 1/2 e to the B and that's going to be X squared. And then we can add our constant on this side Now, since we have that absolute value would be kind of a pain to solve for why, explicitly so I would leave it in this implicit form, but actually, let's call this C one first. Um, I don't wanna have to work with that fraction. So let's just multiply our entire equation by two, so you could just leave it like that. But I just want to slightly rewrite it so I don't have to look at a fraction and multiplying each side by two would give us eat the X squared and then what? Two times a constant, still constant. So this is just another constant which will call C. So this is the solution that I like the most for me not crossed through my two there, but you could have also left it in this form right here. Just kind of depends on which one looks prettier to you.

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