study abroad student needs hints

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coryshap

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I have been going over some problems and have run into some snags. I'm not quite sure how to go about finding the general solution. I am still kind of new to the whole differential equations thing. I am lost on where to begin for four equations. I don't need to know to solve them, rather I was wondering how to get started on them. Like should I use integrating factors, are they separable, do I use substitution... etc. any help would be greatly appreciated. I like figuring them out myself but I am just lost on how to begin and go about solving and am stuck at the moment. :/

1) (xy+y^2) - (x^2)(dy/dx) = 0
2) dy/dx + y = (y^4*)e^x
3) dy/dx + y*tan(x) = sin(x)


Thanks in advance for the help!
 
On the first, the REALLY BIG clue is the degree of each term.

xy, y^2, x^2

All Degree 2. This is just begging for a substitution of the form x = v*y. It should then be separable in v and y.

Let's see what you get.
 
coryshap said:
I have been going over some problems and have run into some snags. I'm not quite sure how to go about finding the general solution. I am still kind of new to the whole differential equations thing. I am lost on where to begin for four equations. I don't need to know to solve them, rather I was wondering how to get started on them. Like should I use integrating factors, are they separable, do I use substitution... etc. any help would be greatly appreciated. I like figuring them out myself but I am just lost on how to begin and go about solving and am stuck at the moment. :/

1) (xy+y^2) - (x^2)(dy/dx) = 0
2) dy/dx + y = (y^4*)e^x
3) dy/dx + y*tan(x) = sin(x)


Thanks in advance for the help!
Click to expand...

3) This problem is of the form:

y' + y *p(x) = g(x)

You'll did need to determine the "integrating factor".

\(\displaystyle e^{\int p . dx } \ = \ e^{\int tan(x) . dx } \ = sec(x)\)

Show us where do you go from here....
 
Ok, here is what I got for the "integrating factor" equation:

using sec(x) you multiply both sides by it to get:
y'sec(x) + ytan(x)*sec(x) = sin(x)sec(x)

which the left side can be changed to make the equation look like:
d[y*sec(x)]/dx = sin(x)*sec(x)

and it can then be put to
y*sec(x) = -log[cos(x)] +c

and using algebra:
y = -log[cos(x)]*cos(x) + cos(x)*(C)

working on the first one now and will post it as soon as I'm done. Thanks for the help
 
So I am stuck on the first one with the substitution. I cant get it to be separable, or at least I don't see how to separate it. Here is what I have so far:

rewrite the original to get dy/dx on one side:
(xy+y^2)/x = dy/dx

then substitute x=vy. This is the part I'm unsure about:
[(vy)y+y^2]/vy = (dy/dv)y + v

Then I threw v to the other side to get:
[(vy)y+y^2-v(vy)]/vy = (dy/dv)y

shouldn't it look different than what I got? I can't get the v's and the y's to cancel and then separate so I can find the general solution. It just seems weird that If I were to even move the dv to the other side that I would be left with a dy*y instead of a dy/y that I could then just flip.
 
coryshap said:
I have been going over some problems and have run into some snags. I'm not quite sure how to go about finding the general solution. I am still kind of new to the whole differential equations thing. I am lost on where to begin for four equations. I don't need to know to solve them, rather I was wondering how to get started on them. Like should I use integrating factors, are they separable, do I use substitution... etc. any help would be greatly appreciated. I like figuring them out myself but I am just lost on how to begin and go about solving and am stuck at the moment. :/

1) (xy+y^2) - (x^2)(dy/dx) = 0

divide by y[sup:3pe8xqyz]2[/sup:3pe8xqyz]

[(x/y) + 1] dx - (x/y)[sup:3pe8xqyz]2[/sup:3pe8xqyz] dy = 0

substitute

v = x/y ? dx = y dv + v dy

[v + 1](y dv + v dy) - v[sup:3pe8xqyz]2[/sup:3pe8xqyz]dy = 0

[v + 1](y dv) + v[sup:3pe8xqyz]2[/sup:3pe8xqyz] dy + vdy - v[sup:3pe8xqyz]2[/sup:3pe8xqyz]dy = 0

[v + 1](y dv) + vdy = 0

[v + 1]/v (dv) = - dy/y

Now continue......

Thanks in advance for the help!
Click to expand...
 
now I see how that one works. I think my biggest problem is just figuring out how to get started in the right direction, or as your sig. suggests just getting used to it.

I got:
c+v+log(v) = -log(y)
with reinserting v=x/y and some rearranging you get the correct answer:
y = x/[c-log(x)]

Im just still stumped on this one:
dy/dx + y = (y^4)*(e^x)
would you do the same sort of substitution to attempt to make it a linear separable equation?

And I have been looking at one that involves no x's as well:
dy/dx = (y+3)(y-5)

for this one Im going at it by separating it to make:
dx = [1/(y^2-2y-15)]*dy
and then integrating:
x=1/8 (log(5-y)-log(y+3))+c
and all thats left is to rearrange it to get y=stuff? just wondering if I did that one correctly or if I broke a rule somewhere.
 
Never mind. I used bernoullis equation and that worked for the other one. :)
 
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