Question

Determine whether the following equation is exact. If it is,then solve it.

Answer 1

\(cos~\theta~dr-(rsin~\theta-e^\theta)~d\theta=0\)

Answer 2

Step 1: Rearrange the eqn
\(cos~\theta~dr+(-rsin~\theta+e^\theta)d\theta=0\)

Answer 3

Step 2 : Determine whether the eqn is exact.
\(\frac{dM}{d\theta}=-sin~\theta~~~,~~~\frac{dN}{dr}=-sin~\theta\)
Since \(\frac{dM}{d\theta}=\frac{dN}{dr}\),this eqn is exact.

Answer 4

Step 3 : find \(u(r,\theta)\)
\(\frac{du}{dr}=0=M(r,\theta)~~~,~~~\frac{du}{d\theta}=-rcos~\theta+e^\theta=N(r,\theta)\)

Answer 5

Then,integrate both sides,
\(u(r,\theta)=X_1(\theta)~~~,~~~u(r,\theta)=-rsin~\theta+e^\theta+X_2(r)\)

Compare \(X_1(\theta)~~~and~~~X_2(r) ~:~\)
\(X_1(\theta)=-rsin~\theta+e^\theta+X_2(r)\)

\(X_1(\theta)=e^\theta\)
\(X_2(r)=0\)

is it correct so far ?

Answer 6

\[\large\rm u_r=M\]Integrating,\[\large\rm u=\int\limits M~dr=~r \cos \theta+g(\theta)\]Looking at the other one,\[\large\rm u_{\theta}=N\]Integrating,\[\large\rm u=\int\limits N~d \theta~=~r \cos \theta + e^{\theta} + h(r)\]

Answer 7

I'm getting a little confused by your fancy x1's and x2's lol
so I wrote it in a way that makes more sense to me x'D

Answer 8

There are two method to solve it
Method 1:
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Method 2:
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I get confused with that r and \(\theta\) maybe we can change it to x and y if u want X'D

Answer 9

Ya I guess I was applying the second method,
See steps (3) and (4)?
I did that and ended up with,\[\large\rm u=r \cos \theta+g(\theta)\]\[\large\rm u=r \cos \theta + e^{\theta} + h(r)\]

So clearly g(theta) = e^theta
and h(r) = 0 because there's nothing from the first integration (involving r) which is missing in the second.\[\large\rm u(r,\theta)=r \cos \theta+e^{\theta}\]

Answer 10

I think that's right :O
Is that confusing?

Answer 11

At the moment no ^

Answer 12

\(\color{#0cbb34}{\text{Originally Posted by}}\) @Zepdrix
Ya I guess I was applying the second method,
See steps (3) and (4)?
I did that and ended up with,\[\large\rm u=r \cos \theta+g(\theta)\]\[\large\rm u=r \cos \theta + e^{\theta} + h(r)\]

So clearly g(theta) = e^theta
and h(r) = 0 because there's nothing from the first integration (involving r) which is missing in the second.\[\large\rm u(r,\theta)=r \cos \theta+e^{\theta}\]
\(\color{#0cbb34}{\text{End of Quote}}\)
owh,can u show me ur previous step?
I think I did something wrong >.<

Answer 13

You mean the integration process?
Because it looks like you got step 2 without any problems

Answer 14

owh,LOL
I know where I went wrong >.<

Answer 15

Step 3
was wrong
*facepalm*
shouldn't hv differentiate it,lmao

Answer 16

Thanks 4 the help, zepdrix ^_^

Answer 17

yay team