Differential Equation. A little help please kind lads

[Seiya]

Differential Equation. A little help please kind lads :)

Hey, I've got this problem... i pay attention in class and read the chapter in the book but i can't seem to know how to solve this? ANy help is greatly appreciated lads... thank you

y'tanx=a+y y(pi/3)=a 0<x<(pi/2)

What i done

(dy/dx)tanx = a + y
dytanx=(a+y)dx
dy/(a+y)=dx/tanx

integral of both sides...

ln(a+y)=-csc^2(x) ? This isn't the answer in the back of the book, anyone can tell me where I've gone wrong? 1/tanx = cot x and the integeral of cotan is -csc^2... oh i also assumed a was a constant, is it not? :S :( thanks guys

 

[dicerandom]

Consider an alternative method of integrating cot(x), try doing a u substitution with u=sin(x).

 

[Seiya]

LN(a+y) = LN (sinx) + C

Like this?

Then

a+y = sinx +c (take e^ of both sides)

so y = sinx - a

the answer is y= (4a*sinx)/sqrt(3) - a

my answer is close to that i suppose now i have to do something with that y(p/3) = a ... any hints?

 

[Seiya]

sorry for the double post ... my internet messed up..im tryin to see how to slve y(pi/3) = a now

 

[dicerandom]

You did the integration correctly, however there's an algebraic mistake when you raised both sides of your equation to the power of e:

$$e^{\ln(\sin{x}) + c} \neq \sin{x} + c$$
$$e^{\ln(\sin{x}) + c} = e^{\ln(\sin{x})} e^{c} = c \sin{x}$$

(relabling e^c as just c, since they're both constants anyway).

Also, when you're matching boundary conditions the general method of attack is to evaluate the relevant function at the value specified then set it equal to the value of the boundary condition.

 

[Seiya]

so i assume that i did a mistake on the a+y side as well?

 

[dicerandom]

so i assume that i did a mistake on the a+y side as well?

No, that side's fine since the a+y is all within the natural log.

 

[Seiya]

ok so i have

y=e^c sinx - a

now i have to do y(pi/3) = e^c*sin(pi/3)-a =a and solve for c?

*try*

 

[Seiya]

thanks a lot i got it now, i really appreciate it :) thank you so much