Solve Diff Eqns for Polar Functions & Critical Points

[vinverth]

Hello everyone..Find it embarrassing enough on asking a question on my very first post but I've been an avid reader of the forums for the past couple of months and been finding what i need for all my assignments here.So a big Thank You to all who've helped.I'm A EE grad and have a math course in my final semester so am a complete noob when it comes to grad math courses,a little consideration here while posting replies or even answers.So here i have a couple of q's whose answers or at least a decent start I've been searching all over the web.

1.Find the differential equations for the polar functions r,ө of the following two-dimensional systems.

(a) x'=x+y
y'=x-y

2.Locate the critical points of the following systems.

(a) x'=x-y²
y'=x²-y²
These are both separate questions.Answers to anyone pleasezzz..
(b) x'=sin(y)
y'=cos(x)Thank You again to everyone and please bail me out guys!

 

[fzero]

1.Find the differential equations for the polar functions r,ө of the following two-dimensional systems.

(a) x'=x+y
y'=x-y

Polar coordinates are defined by

$$x = r \cos\theta,~y=r\sin\theta.$$

You should rewrite the equations using these and solve for $$r', \theta'$$.

2.Locate the critical points of the following systems.

(a) x'=x-y²
y'=x²-y²
These are both separate questions.Answers to anyone pleasezzz..
(b) x'=sin(y)
y'=cos(x)

Critical points are the points where the derivatives of functions either don't exist or are zero.

 

[vinverth]

I already tried that approach but it got me nowhere..what confuses me is the fact that the equations are already in their first derived form for both q's wrt x and y.
i tried to plug in x=rcosө and y=rsinө which gives me x'=rcosө+rsinө.
(rcosө)'=rcosө+rsinө
How do i proceed frm here..

and for the second q..since x'=sin(y) and y'=cos(x)..do i just check for what values x' and y' are 0 and figure out their critical points.
Any kind of help or a start is appreciated..and I've been trying to solve them in every possible way via research on internet.but maybe its just my fundamentals..too weak at them..like i said before i hail from a diff background

 

[fzero]

I already tried that approach but it got me nowhere..what confuses me is the fact that the equations are already in their first derived form for both q's wrt x and y.
i tried to plug in x=rcosө and y=rsinө which gives me x'=rcosө+rsinө.
(rcosө)'=rcosө+rsinө
How do i proceed frm here..

Use the product rule for derivatives

$$(fg)' = f' g + f g'.$$

and for the second q..since x'=sin(y) and y'=cos(x)..do i just check for what values x' and y' are 0 and figure out their critical points.

The (x,y) values for which either x'=0 or y'=0 are critical points.