initial value problem and critical value of a0.

• Apr 5th 2011, 02:44 AM
Taurus3
initial value problem and critical value of a0.

Given: (sint)y' + (cost)y=e^t; y(1)=a, 0<t<pi

a. find the initial value problem and find the critical value a0 exactly. '
so to make this into standard form, I divided by sint.
so: y' + (cost)y/(sint)=e^t/(sint)
μ(t)= e^(ln(sint))
Multiplying both sides by e^(ln(sint)): e^(ln(sint))y' + cost/sint*e^(ln(sint))y=e^t/sint*e^(ln(sint)). Then I got stuck....8(

b. describe the behavior of the solution corresponding to the initial value a0.

c. Draw the direction field of the given differential equation. How do the solutions behave as t goes to 0? Does the behavior depend of the choice of the initial value a? Why or why not? Let a0 be the value for a for which the transition from one type of behavior to another occurs. Estimate the value of a0.
• Apr 5th 2011, 04:09 AM
amul28
Try using the Integrating Factor method.

$\frac{dy}{dt}+ycot(t)=\frac{e^t}{sin(t)}$
• Apr 5th 2011, 04:46 AM
Ackbeet
Or even quicker: notice that the LHS is the total derivative

$\dfrac{d}{dt}(\sin(t)y).$

Then integrate immediately. Essentially, this is what your integrating factor method is going to do anyway. That's what the integrating factor method is designed to do.
• Apr 5th 2011, 04:50 AM
amul28
i din't get the method you specified?
• Apr 5th 2011, 04:51 AM
Ackbeet
Quote:

Originally Posted by amul28
i din't get the method you specified?

• Apr 5th 2011, 05:07 AM
amul28
Quote:

Originally Posted by Ackbeet

sorry, i am replying to you.
• Apr 5th 2011, 05:17 AM
Ackbeet
Well, if you follow the program I've laid out for you, you will obtain

$\dfrac{d}{dt}(\sin(t)y)=e^{t}.$ To see why this is equivalent to the original DE, just take the derivative on the LHS to get

$\sin(t)\dot{y}+\cos(t)y.$

Hence, if you integrate with respect to time, you get

$\displaystyle\int\frac{d}{dt}(\sin(t)y)\,dt=\int e^{t}\,dt,$ or, by using the Fundamental Theorem of the Calculus, you get

$\sin(t)y=e^{t}+C.$ Can you continue from here? Is everything clear now?
• Apr 5th 2011, 11:31 AM
Taurus3
Can you please solve the whole thing? I really don't get it.
• Apr 6th 2011, 02:14 AM
Ackbeet
Quote:

Originally Posted by Taurus3
Can you please solve the whole thing? I really don't get it.

That isn't the way this forum operates. I've essentially solved the DE for you. The business about critical points is something for which you'll need to review your notes, the definition of critical point, etc.