Question

@ikram002p

Answer 1

check if below DE is exact and solve using a simple line integral

\[\large (2xy-9x^2)dx+(2y+x^2 + 1)dy\]

Answer 2

These type of questions must be censored :P

Answer 3

ok ill try

M dx+N dy
is it =0 ?

Answer 4

not gonna happen since it already exists in paul's page :P
http://tutorial.math.lamar.edu/Classes/DE/Exact.aspx

Answer 5

:D

Answer 6

yes !

Answer 7

yes zero ?

Answer 8

\[\large (2xy-9x^2)dx+(2y+x^2 + 1)dy \]

you want to read it as

\[\large M dx+Ndy = 0\]

is it ?

Answer 9

ok ok
so
\(N_x=2x\)
\(M_y=2x\)
then exact

Answer 10

yep !

Answer 11

nw need tofind work done by simple path like this

Answer 12

on simple path not done by lol

Answer 13

btw , thanks for teaching me this :)

Answer 14

yes :) looks good, keep going..
may be define your vector field and the path first - just for clarity

Answer 15

so ,
on \( C_1 \)
y=0
dy=0
work= \(\int_0^x -9 x^2\) dx

Answer 16

so its -3 x^3

Answer 17

yes !!

Answer 18

work on c2
x=0
dx=0
work=\(\int_0^y 2y+1\)

so its y^2+y

Answer 19

careful, on C2, x is NOT 0
x = \(x_1\) = constant

Answer 20

work on c2
x=\(x_1\)
dx=0

Answer 21

ohh sorry my bad

Answer 22

setup ur integral again for C2

Answer 23

then its

\(y^2 +( x_1 +1 ) y\)

Answer 24

Looks perfect!
so whats the solution to the DE

Answer 25

F=\(-3x^3+y^2+(x_1+1)y\)

Answer 26

could we add constant to this ?

Answer 27

call it potential funciton

Answer 28

ohkk

Answer 29

yes the solution is :

C1 + C2 = k

Answer 30

thx :)

Answer 31

np :) wana try one more ?

Answer 32

ok sure

Answer 33

example3 from paul
http://tutorial.math.lamar.edu/Classes/DE/Exact.aspx

Answer 34

solve \[\large 2xy^2 + 4 = 2(3-x^2y)y'\]

Answer 35

im going for early lunch... wil check once im back... good luck :)

Answer 36

\((2x y^2+4) dx-2(3-x^2y) dy=0\)
exact check
sol :-
\(P=\int_0^x (2xy^2+4) dx + \int_0^y (-6+x_1^2y)dy\)
\(y=2/3 x^3 y +4x -6y +1/2 x_1^2 y^2 +k \)
then
y(-1)=8
we could know something with it lol

Answer 37

ok have a nice lunch !

Answer 38

forget about the initial conditions,
just solve the DE..

Answer 39

OMG! looks you have solved it already xD

Answer 40

:)

Answer 41

So ikram do you need another, is that what this means?

Answer 42

sure , wanna learn all about DE

Answer 43

one sec let me make one up for you that's similar to what you've been doing. I just need to take a couple derivatives. =)

Answer 44

huh ohk! make it easy lol

Answer 45

Sorry give me more time, I didn't want to give you that one, I realized it was too difficult since it was a system of 3 exact equations.

Answer 46

O.O
ohkk !
well when i done with two equation
teach me how to do it with 3

Answer 47

Well do you know what the gradient of a scalar field is or the curl of a vector is?

Answer 48

Also, do you know how to use an integrating factor to solve an exact equation? Maybe that would be a good idea for your next one. =)

Answer 49

This might be a good one to try. See if it's exact after you put it into a form that's homogeneous. \[\LARGE \frac{dy}{dx}=-\frac{3xy+y^2}{x^2+xy}\]

Answer 50

wait before i start , i still learn seperable equation xD

Answer 51

This is an exact equation, it just looks separable.

Answer 52

ok teach me how to do this one
and give me Hw xD

Answer 53

Haha ok, well first get it into the form that you're used to by multiplying the denominator by both sides, then subtract one thing from one side of the equation to get that form you're used to.

Mdx+Ndy=0

I think that's what it is.

Answer 54

ok then ?

Answer 55

Except this time check to see that \[\Large M_y \ne N_x\]

Answer 56

So we must figure out how to get past this. So we use an integrating factor, which we multiply by the equation Mdx+Ndy=0 so that we can make it exact.

Answer 57

ok so try to find a function that if we myltiply with both M dx +N dy =0
then we got
\(N_x=M_y\)

Answer 58

\[\Large \mu (x,y)Mdx+\mu (x,y)Ndy=0\] Since all we're really doing is multiplying by zero, it should be safe. Now let's go through the same process of checking for exactness with this factor and MAKE it exact.

Answer 59

Yeah you got it.

Answer 60

ok good :)
continue plz

Answer 61

So what did you get for your integrating factor?

Answer 62

wait lemme think

Answer 63

ok u tell T_T

Answer 64

Ok haha I think I should have written mu(x) not mu(x,y) because our integrating factor only needs to be a function of one of our variables. To make two things line up, we only need to change one thing, right?

@ikram002p what are you getting for M and N from the function earlier?

Answer 65

well
\(x u_x - 3x u_y =0\)
xD
then \(u_y =0 \) ?

Answer 66

Sorry I almost fell asleep at my desk, I think I should probably go. This is the example I found, if you want to continue it. I'm not a fan of doing it by "the formula" because it requires me to remember stuff. It's better to just solve it by understanding in my opinion.

http://www.sosmath.com/diffeq/first/intfactor/Example/Example.html

Answer 67

okkk :)
thank you