SOLUTION: Solve the differential equation: 3e^xtanydx+(1-e^x)sec^2ydy=0

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Question 637388: Solve the differential equation:
3e^xtanydx+(1-e^x)sec^2ydy=0
Answer by Edwin McCravy(20060) About Me  (Show Source):
You can put this solution on YOUR website!
3e%5Ex%2Atan%28y%29dx + %281-e%5Ex%29%2Asec%5E2%28y%29dy = 0

Separate the variables by dividing through by %281-e%5Ex%29tan%28y%29%0D%0A

%283e%5Ex%2Atan%28y%29dx%29%2F%28%281-e%5Ex%29tan%28y%29%29 + %28%281-e%5Ex%29%2Asec%5E2%28y%29dy%29%2F%28%281-e%5Ex%29tan%28y%29%29 = 0%2F%28%281-e%5Ex%29tan%28y%29%29

%283e%5Ex%2Across%28tan%28y%29%29dx%29%2F%28%281-e%5Ex%29cross%28tan%28y%29%29%29 + %28%28cross%281-e%5Ex%29%29%2Asec%5E2%28y%29dy%29%2F%28%28cross%281-e%5Ex%29%29tan%28y%29%29 = 0

%283e%5Ex%2Adx%29%2F%281-e%5Ex%29 + %28sec%5E2%28y%29dy%29%2Ftan%28y%29 = 0

Now we integrate all three terms:

int%28%283e%5Ex%29dx%2F%281-e%5Ex%29%29%29 + int%28%28sec%5E2%28y%29dy%29%2Ftan%28y%29%29= int%280%29

Both terms on the left can be integrated using the formula int%28du%2Fu%29 = ln|u|+C
The second integral is already set up for that. We take the 3 out
of the integral on the first term:

3int%28%28e%5Ex%29dx%2F%281-e%5Ex%29%29%29 + int%28%28sec%5E2%28y%29dy%29%2Ftan%28y%29%29= int%280%29

We need a negative sign in the numerator of the first fraction so it will
be in the form:

-3int%28%28-e%5Ex%29dx%2F%281-e%5Ex%29%29%29 + int%28%28sec%5E2%28y%29dy%29%2Ftan%28y%29%29= int%280%29

Integrating using ln(C) for the arbitrary constant so everything will
be natural logs:

-3·ln|1-ex| + ln|tan(y)| = ln(C)

               ln|tan(y)| = 3·ln|1-ex| + ln(C)

Use a rule of logs to move the 3 to an exponent:

               ln|tan(y)| = ln|1-ex|3 + ln(C)

Write the right side as the natural log of a product:

               ln|tan(y)| = ln[C|1-ex|3]

Take anti-logs of both sides:

               |tan(y)| = C|1-ex|3

Since the constant C can be positive or negative, we don't need
the absolute values:

               tan(y) = C(1-ex)3

And if we like we can solve for y by taking arctangents of both sides:

                    y = arctan[C(1-ex)3]

Edwin