Differential Equation

[earnest smelt]

This is my attempt at solving this differential equation, I think it’s separable, but I don’t know if my approach is correct. Any help would be appreciated.

[tame frost]

What is the main question?

/ do you mean
$d/dx ( $√y $cos²$($√y)

[nimble sedge]

💚_MATHS_💚
Compile Error! Click the reaction for more information.
(You may edit your message to recompile.)

[bronze stag]

Make an appropriate substitution for the left integral, which should be obvious, but if not, try ||u=\sqrt{y}||

[bronze stag]

Gonna also roast your handwriting a tiny bit be careful there that it doesn’t get misread!

[earnest smelt]

Lmao please don’t

It’s a differential equation

[earnest smelt]

This is the question

The first image

[bronze stag]

Haha don’t worry, I managed to make it out(!) + mine isn’t much better either to begin with

[bronze stag]

I’ll reiterate this though, up to what you did is the right track

[earnest smelt]

Okay

Here’s the problem

The integral on the left hand side

Do I do u sub for sqrt(y) and do integration by parts?

Or is it simpler that that?

[bronze stag]

Yes, much simpler - how well do you know your trig functions?

[earnest smelt]

Well

[bronze stag]

What you should get is something that, ideally, is easily identifiable as a derivative of something…

[earnest smelt]

Sec^2(x) is the derivative of tanx if I’m not mistaken

[bronze stag]

There you go that should get you there, does it?

[earnest smelt]

Okay so so far I have 1/sqrt(y) • sec^2(x)

Idk how do I get the tanx tho

[bronze stag]

Did you substitute properly?

Check the “du” you should have

[bronze stag]

(Also gonna be that annoying guy, but careful of using x, seeing there’s another x on the other side too!)

Can’t have people losing marks now

[earnest smelt]

Oh yeah about that-

I think I got it

u = sqrt(y)

The expression turns into 2/cos^2(u)

Which can be written as 2sec^2(x)

u*

THEN I get the 2tan(u)

= 2tan(sqrt(y)) +C

[bronze stag]

Yep! (Though you don’t need a +c because you have one on the other side!)

[earnest smelt]

Obviously the constant doesn’t have to be on both sides

[earnest smelt]

Exactlyyyyyy

[bronze stag]

Got’eem

But yep, there we go, you got it 💪

[earnest smelt]

I still need to solve for y lol

That’s easy tho

Atleast that’s what I think

[bronze stag]

Yea would say that that’s pretty easy, think you’ll survive with rearranging that!

[earnest smelt]

Yess

I got a quiz tmro so I’m gonna need more practice😭

[bronze stag]

Damn so early, not even into the new year and they’re already testing you best of luck for it tho, sure you’ll do well

But otherwise, is all of that all good?

[earnest smelt]

Yeah sadly

[earnest smelt]

Yeppers!!

I thank you so much for your help

[bronze stag]

No worries, been a pleasure! Happy to help

We can mark this solved then?

[earnest smelt]

You’re a hero🫡🫡

[earnest smelt]

Yes ofc

How tho

[bronze stag]

Awww thank you very much! again best of luck with the practice and test!

.solved

[dim schooner]

Solved

Post marked as solved by @bronze stag.

Use .unsolved if this was a mistake.

[bronze stag]

Marks it as solved

[earnest smelt]

.solved

/.solved

Lmao

I’m so unfamiliar with this

[bronze stag]

I’ve already marked it haha

[earnest smelt]

Oh

😭😭😭😭

[bronze stag]

But that’s the command for it, and you can use the .unsolved to reopen it (also the .close and .reopen still work in here too!)

Anyways, take care, have a good one and enjoy as much of the new year as you can!