Linear First Order D.E

1. May 11, 2005

splitendz

Solve y’ – y*tan[x] = 2sin[x].

I keep arriving at the answer: y = 2ln(sec[x]) / sec[x] - c/sec[x]
for this question.

According to my textbook the correct answer is: y = (c - cos[x]) * sec[x]. Can anyone explain how to obtain this answer?

Thanks :)

2. May 11, 2005

dextercioby

Show us your work.Because of that natural logarithm,the two answers are not equivalent (equal for some value of the integration constant),therefore at least one of them is wrong.

Daniel.

3. May 11, 2005

HallsofIvy

Staff Emeritus
The text answer is correct. I suspect you overlooked the fact that integrating will give you a ln(y) as well as "ln(sec(x))".

Since this is a linear equation, you can do the homogeneous and non-homogeneous parts separately. The associated homogeneous equation is just y'- y tan x= 0 or
y'= y tan x so dy/y= tan x dx= (sin x dx)/cos x.
To find a solution to the non-homogeneous part you could use "variation of parameters": try a solutution of the form y(x)= u(x) sec(x) (I'm giving away part of the answer there!), differentiate and plug into the equation to find a simple equation for u(x).
Or just guess a simple solution!

4. May 11, 2005

splitendz

Ah right. Thanks a lot for your help HallsofIvy! I've got it now.