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)   (Show Source): You can put this solution on YOUR website!
 +  = 0

Separate the variables by dividing through by 

 +  = 

 +  = 0

 +  = 0

Now we integrate all three terms:

 + = 

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

 + = 

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

 + = 

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

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