Question

What is the solution of the following DIFFERENTIAL EQUATION for the initial value problem: y(0) = -2

Eqn: y' + (tanx)y = 2(cosx)^2

Answer 1

\[\frac{dy}{dx} + tanx(y) = 2\cos^2x\]

Answer 2

@experimentX

Answer 3

@apoorvk

Answer 4

@Callisto

Answer 5

you mean tanx multiplied by y - right?

Answer 6

Yes

Answer 7

Use the funda of Integrating factor - know it?

Answer 8

I am trying to get the form:
\[f(x)dx = f(y)dy\]

Answer 9

first what kind of equation is that?? and how do you solve it??

Answer 10

This is (I think) a first order seperable differential equation. You solve by having:
\[f(x)dx = f(y)dy \]

or

Answer 11

Because you have to integrate both sides.

Answer 12

SO you have to attach the y terms with dy, x terms with dx.

Answer 13

it's called linear first order differential equation!!

Answer 14

Umm... you don't need that really.

If a differential equation is of the form:

\[\frac{dy}{dx} + Py = Q\]

where both 'P' and 'Q' are functions of 'x', then,

\[y.e^{\int P.dx}=\int Q.e^{\int P.dx}.dx ~+~k\]


this is called a linear differential equation.

Answer 15

\[Here,~e^{\int P.dx}~is~known~as~the~INTEGRATING~FACTOR.\]

Answer 16

I think I am going to read further about that. Right now, I would like you to help me with a question that I have already read about:

\[\int\limits_{\pi/2}^{\pi} 34\csc(x)dx\]

Is this convergent or divergent?

Answer 17

I have got up to this part:
\[\lim_{a \rightarrow \pi-} \ln|\csc(a) - \cot(a)|\]

I then end up evaluating it and I get 0 for the argument in the ln.

Answer 18

\[\int\limits\limits_{\pi/2}^{\pi} 34\csc(x)dx
= \lim_{a \rightarrow \pi-} \int\limits\limits_{\pi/2}^{a} 34\csc(x)dx\]

Answer 19

I know the integral of \[\csc(x)\]

Answer 20

\[\int\limits_{}^{}\csc(x) dx = \ln|\csc(x) - \cot(x)| + C\]

Answer 21

I think I can easily show that this integral is in fact convergent by the comparison theorem. However, I am having trouble showing this from the integral in question itself.

Answer 22

@shivam_bhalla
@amistre64
@satellite73
@FoolForMath
@Zarkon
@ParthKohli

Answer 23

@TuringTest
@Mertsj
@Ishaan94

Answer 24

try asking wolf

Answer 25

No thanks. I need to show a fully justified solution. I KNOW that it converges, but I do not KNOW how to show it because I keep getting 0 in the argument of the natural log

Answer 26

@Ishaan94 we are discussing not the differential equation, but the convergence question later on

Answer 27

Oh Lool I was typing the solution to differential equation.

Answer 28

http://www.wolframalpha.com/input/?i=y%27+%2B+%28tanx%29y+%3D+2%28cosx%29^2%2C+y%280%29+%3D+-2

Answer 29

@Mertsj

Answer 30

@AccessDenied
@FoolForMath
@apoorvk
@eliassaab

Answer 31

\[ \int \tan x dx = - \ln |\cos x|\]
Integrating Factor is
\[ e^{-\ln|\cos x|} = \sec x\]

your solution is
\[ y = (\sec x) ^{- 1} \int \sec x 2 \cos^2x dx + C(\sec x)^{-1}\]
\[ y = 2\cos x \int \cos x dx + C\cos x\]
\[ y = 2\cos x (-\sin x) + C\cos x\]
You know how to solve from here right??

Answer 32

Sorry I was talking about the convergence question.

Answer 33

Maybe I should post a new question for that. Hold on

Answer 34

Oh ... LOL
i guess you should ... i got confused