Solving First Order ODEs: Tips & Tricks

[STEMucator]

Homework Statement

Couple of ODE's I'm having trouble with, a bit rusty. They're all first orders.

(1) y' - y = -y³
(2) http://gyazo.com/4a83c6f72c552d1679b9bf95f644599c

Homework Equations

Substitution, integrating factors.

The Attempt at a Solution

I'm not quite sure how to go about solving the first one, integrating factors don't work, separation is obviously not happening. I highly doubt that advanced methods like the method of successive approximations is needed. Any pointers on this one would be nice.

The second one is tricky, I'm trying to turn it into a homogeneous equation so I can make the substitution v = y/x and go from there. I'm just having a bit of trouble putting it in the required form. I tried dividing by a few things like xy, x, etc, but to no avail.

Why are first orders harder than nth orders...

 

[Dick]

Homework Statement

Couple of ODE's I'm having trouble with, a bit rusty. They're all first orders.

(1) y' - y = -y³
(2) http://gyazo.com/4a83c6f72c552d1679b9bf95f644599c

Homework Equations

Substitution, integrating factors.

The Attempt at a Solution

I'm not quite sure how to go about solving the first one, integrating factors don't work, separation is obviously not happening. I highly doubt that advanced methods like the method of successive approximations is needed. Any pointers on this one would be nice.

The second one is tricky, I'm trying to turn it into a homogeneous equation so I can make the substitution v = y/x and go from there. I'm just having a bit of trouble putting it in the required form. I tried dividing by a few things like xy, x, etc, but to no avail.

Why are first orders harder than nth orders...

Start with the first one. Why do you think separation isn't happening. It's happening for me.

 

[STEMucator]

Start with the first one. Why do you think separation isn't happening. It's happening for me.

Wow, that's what I get for being sloppy. Just added y to both sides then divided through. Easy stuff. Simple partial fraction afterwards.

The second one though?

 

[Dick]

Wow, that's what I get for being sloppy. Just added y to both sides then divided through. Easy stuff. Simple partial fraction afterwards.

The second one though?

You are missing the easy stuff. The right side is a function of y/x. Maybe try the substitution z=y/x.

 

[STEMucator]

You are missing the easy stuff. The right side is a function of y/x. Maybe try the substitution z=y/x.

I realized I could separate it into :

$1 + \frac{y}{x} + (\frac{y}{x})^{2}$

The rest is... straightforward to say the least.

Thank you very much for the refresher, sometimes the easiest things are also easy to forget.

 

[Dick]

I realized I could separate it into :

$1 + \frac{y}{x} + (\frac{y}{x})^{2}$

The rest is... straightforward to say the least.

Thank you very much for the refresher, sometimes the easiest things are also easy to forget.

No problem. Actually nth orderers are much harder. It just seems that way because they are so much harder they are usually stuck with giving easier examples.

 

[STEMucator]

No problem. Actually nth orderers are much harder. It just seems that way because they are so much harder they are usually stuck with giving easier examples.

I feel with nth orders and systems of eqs there is a wide variety of well defined methods to solve a wide class of problems. It becomes systematic after second orders, i.e variation, undetermined, laplace, etc.

First orders slipped my mind pretty hard.