Calculus Help Please!!?

Question

Working on old AP Calc BC practice tests, I ran across this, and no one in my class knows how to solve it:

a) In the interval 0
(cot x) (dy/dx) + y = (csc x)

b)Find the solution of hte differential equation in part a that satisfies the condition that y = 0 when x = pi/3



Is it a separable equation? If so, how do I separate the terms??

If not, please help!! Thanks!!

Answer 1

a) cot x dy/dx + y = csc x
<=> (cos x / sin x) dy/dx + y = 1/sin x
<=> dy/dx + (sin x / cos x) y = 1/cos x
This is linear, so we have an integrating factor of e^∫(sin x / cos x) dx
Now ∫(sin x / cos x) dx = - ln |cos x| + c
So the integrating factor is e^(-ln cos x) = 1/cos x (note |cos x| = cos x on the given interval).

So we get
(1/cos x) dy/dx + (sin x / cos^2 x) y = 1/cos^2 x
<=> d/dx (y/cos x) = 1/cos^2 x
<=> y/cos x = ∫(sec^2 x dx)
<=> y = cos x (tan x + c)
<=> y = sin x + c cos x.

b) y(π/3) = sin π/3 + c cos π/3 = 0
=> √3 / 2 + c / 2 = 0
=> c = -√3
So y = sin x - √3 cos x.

Answer 2

cot (x) y' + y = csc(x)
(cos(x) / sin(x)) y' + y = (1/sin(x))
cos(x) y' + sin(x) y = 1

It's not seperable, but you can look for a solution in the form of y = a sin(x) + b cos(x). This means dy/dx = a cos(x) - b sin(x). Plug these into the differential equation:

cos(x) (a cos(x) - b sin(x)) + sin(x) (a sin(x) + b cos(x)) = 1
a cos²(x) - b sin(x)cos(x) + a sin²(x) + b cos(x)sin(x) = 1
a (cos²(x) + sin²(x)) = 1
a = 1. It seems that b can be anything, so the solution is
y = sin(x) + b cos(x) for constant b.

Plug in the initial condition:
0 = sin(pi/3) + b cos(pi/3)
0 = 1/2 + b (√3/2), so b = -√3.
y = sin(x) - √3cos(x)

Answer 3

I don't believe that you can use separation of terms, but you can integrate factors.

(cot x)(dy/dx)+y=csc x <--can also be written as

(dy/dx) + y (tan x)= sec x <--if you divide everything by cot x

Now we can say that
p(x)=tan x and q(x)=sec x
Integrate p(x) to get P(x)= -ln |cos(x)| + c
u (greek)=e^( -ln |cos(x)| + c )

To integrate factors:
y=(1/u) S uq(x) dx

So,
y=(e^(ln |cos(x)| + c)) S {[e^( -ln |cos(x)| + c )]sec x} dx

Have fun taking that integral.

I hope that this helps.

Answer 4

it can be solved as a linear equation

try the following link

Answer 5

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