Differential equation

[Roark]

Hi,

 

Here's the problem: dy/dx = y/a

 

The answer is y=ce^(x/a)

 

I can get to : x = a ln |y| + c and I know that x=ln y is the same as y = e^x but I can't make the connection to the answer.

 

Thanks.

[Dave]

Originally posted by Roark

dy/dx = y/a

 

i'll do the entire thing (mainly through boredom )

 

you need to start by seperating the variables, which isn't all that hard:

 

:lint: (1/y)dy = :lint: (1/a)dx

therefore ln |y| = x/a + const.

 

now, since this is a constant we're talking about here, we can do more or less anything with it. since we have ln's floating about, it would be sensible to have a constant involving a natural log, since this would still be a constant and it doesn't really do anything wrong. after all, if you had any values of x and y at different points, you'd still get the same answer for the constant.

 

so:

 

ln (y) = x/a + ln c

 

therefore ln (y/c) = x/a

y=ce^(x/a) as required.

 

these types of questions are a bit misleading, because you never really know the form that the answer is going to be in. as a general rule, when i integrate something that has natural logs floating around, i tend to use a log constant unless i already know what the answer is going to be. you'll also find that a lot of the time, you'll be getting different values of x and y to find the constant.

[Roark]

Thanks! I actually got it last night.

 

It seems like cheating when you put c = ln c. I know c can be anything but isn't the answer now only true if c = ln c?

[blike]

Dave, you have a degree in a field of math?

[Dave]

Originally posted by Roark

Thanks! I actually got it last night.

 

It seems like cheating when you put c = ln c. I know c can be anything but isn't the answer now only true if c = ln c?

 

in a way i suppose it is cheating, but the answer previously obtained still stands. that is a solution to the differential equation stated, they just wanted to see that it can be written in it's most simplest form if you let the constant of integration be written in a logarithmic form.

 

as for the degree, i'm currently doing my A2 levels at school/college in preparation for a maths degree at warwick or york, whichever i get into. my current plan is to do the 4 year MMath course and then go on to do my PhD in maybe some pure or applied field of maths. (i hope )