Separation of variables

[SilverKing]

Hi everyone,

I've the following problem:

Determine the general solution and the particular solution of e^(x) cos(y) dy + (1+e^(x)) sin(y) dx = 0 when y(0)=0

Solution:

Separating the variables:

e^(x)/(1+e^(x)) dx + sin(y) / cos(y) dy = 0

Integrating:

ln|1+e^(x)| - ln|cos(y)| = ln |c| *

ln |1+e^(x)/cos(y) = ln|c|

1+e^(x) = c|cos(y)| ---> The general solution

Substituting x=0, y=0:

c=2

Which leads to:

1+e^(x) = 2|cos(y)| ---> The particular solution

I've understood it all, but what confusing me is the equation *. Why the integration of 0 is ln c and not just c?

 

[Deleted member 4993]

Hi everyone,

I've the following problem:




I've understood it all, but what confusing me is the equation *. Why the integration of 0 is ln c and not just c?

That is done for convenience.

ln(C₁) = c = another constant

 

[SilverKing]

But that's will affects on the final answer. So, can I assume anything for the arbitrary constant that would help me simplifying the final answer?

 

[Deleted member 4993]

But that's will affects on the final answer. So, can I assume anything for the arbitrary constant that would help me simplifying the final answer?

Yes - as long as it is a constant - e^c, sin(c), ln(c), k^c, c^k, c+k, c*k, c/k - everything is a fair game as long as it does not contain a variable,

 

[HallsofIvy]

You have \(\displaystyle e^x cos(y)dy+ (1+ e^x)sin(y)dx= 0\) and then \(\displaystyle \frac{e^x}{1+ e^x}dx+ \frac{sin(y)}{cos(y)}dy= 0\).

The second equation does NOT follow from the first. You have the fractions upside down.

Dividing through by \(\displaystyle e^x\), \(\displaystyle cos(y)dy+ \frac{1+ e^x}{e^x}sin(y)dx= 0\). Dividing through by sin(y), \(\displaystyle \frac{cos(y)}{sin(y)}dy+\frac{1+ e^x}{e^x}dx= 0\).

IF the original equation \(\displaystyle e^x cos(y)dx+ (1+ e^x)sin(y)dy= 0\) THEN what you have would be correct.

(Note that the locations of dy and dx have switched)