Question

differential equation help *question attached below* will give medal

Answer 1

So, I am supposed to solve this differential equation using the substitution given but I've no clue how to start. Can someone walk me through, please?

Answer 2

start by actually making the sub, ie whenever you can fit in the "dy/dx = p " sub, do that...

Answer 3

Okay, then:
y(p')=p^2

Answer 4

so we agree that:

y•dp/dx = p^2 [this is best seen in Leibnitz notation]

what is y•dp/dx•dy/dy ?? ie the LHS of your equation

Answer 5

I'm not seeing the stuff your typed :/

Answer 6

refresh screen

Answer 7

Hmm, now I'm lost.

Answer 8

i am trying not to give you a tsraight answer as that would be pointless. however, not that we took
dp/dx
and we multiplies it by
dy/dy {ie 1}
and we got
y•dp/dx•dy/dy
which can be re-arranged in terms of p and y

Answer 9

clue:

1/2 * 3/3 = 1/3 * 3/2

Answer 10

so
y•dp/dx•dy/dy = ??????

once you get it, the problem is solved.....

Answer 11

I'm still not getting it though :(

Answer 12

But let me still try to process everyhting...

Answer 13

you will do.

1) look at : dp/dx * dy/dy

2) then look at : 1/2 * 3/3 = 1/3 * 3/2

3) can you re-arrange 1) ????

and pls forgive my terrible typing skills.....

Answer 14

He is trying to write your thingy in terms of 2 variables instead of 3.
hint: you will have to use your substitution again

Answer 15

Hm, let me see then.

Answer 16

@iTiaax look at what he labeled 1) and rewrite it (using his example in 2) so you can use your substitution again?

Answer 17

\[\frac{a}{c} \cdot \frac{b}{d}=\frac{b}{c} \cdot \frac{a}{d}\]

Answer 18

multiplication is commutative

Answer 19

\[\frac{dp}{dx} \cdot \frac{dy}{dy}=?\]

Answer 20

dp/dx?

Answer 21

well yeah that is equal to dp/dx
but that isn't what we are looking for exactly

Answer 22

remember you have:
y(p')=p^2
\[y \frac{dp}{dx}=p^2 \]
this is in terms of x,y,p
we want just y,p

Answer 23

we know dy/dy=1
so \[\frac{dp}{dx} \cdot \frac{dy}{dy}=?\]
tried to rewrite this so this is in terms of just y and p

Answer 24

using your substitution p=dy/dx

Answer 25

1/2 * 3/3 = 1/2 ; that is true

but also

1/2 * 3/3 = 1/3 * 3/2 ; that is interesting in calculus because calculus is all about differences.

Answer 26

dp/dx = dp/dy x p

Answer 27

and your x is a times operation right?

Answer 28

well one of them i mean

Answer 29

Oh God, this is setting my brain on fire

Answer 30

has your brain's fires got put out yet?

Answer 31

I have this so far:

yp' = p^2
using the chain rule we can write:
dp/dx = dp/dy * dy/dx
or:
dp/dx= dp/dy
finally, we get:
p' = dp/dy * p
now using the rewritten ODE, we get:
p^2/ y = p* (dp/dy)
or:
p/y = dp/dy

Answer 32

\[\frac{dp}{dx}=\frac{dp}{dy} \cdot \frac{dy}{dx} \\ \frac{dp}{dx}=\frac{dp}{dy} \cdot p \text{ by your subsittuion }\]

Answer 33

\[y \frac{dp}{dx}=p^2 \\ y \frac{dp}{dx} \frac{dy}{dy}=p^2 \\ y \frac{dy}{dx}\frac{dp}{dy}=p^2 \text{ since } dp \cdot dy=dy \cdot dp\]
and we know
\[\frac{dy}{dx}=p\]

your equation should be
\[y \cdot p \cdot \frac{dp}{dy}=p^2 \\ \text{ assuming } p \neq 0 \\ y \cdot \frac{dp}{dy}=p\]
now just use separation of variables to solve

Answer 34

oh I guess that is what you had
I just didn't understand your steps

Answer 35

because your final equation is equivalent to what i have

Answer 36

Do I need to solve for p now?

Answer 37

separate your variables
then integrate both sides

Answer 38

Hmm, let me do it.

Answer 39

p=cy

Answer 40

sounds really really cute

Answer 41

ok one last step

Answer 42

lol

Answer 43

remember we use a substitution
y'=p
\[p=cy \\ y'=cy\]

Answer 44

now you have to solve that last diff equation

Answer 45

So I need to solve y'=cy?

Answer 46

yes

Answer 47

it isn't too bad
it is just another separation of variables

Answer 48

i mean you can also solve it other ways
but the easiest way is the separation of variables way in my opinion

Answer 49

great work @iTiaax and @freckles

Answer 50

Alrighty. Thank you