Thread: Variation of Parameters

I have a test coming up involving variation of parameters and I am practicing. I got stuck on this one, I know it should just be algebra but now I'm lost.

Using the system of equations:

U1'(y1)+U2'(y2) = 0
U1'(y1')+U2'(y2') = f(x)

I need to solve for U1' and U2'.

I started with y" + 16y = 4Tan(4t)

I found that r = 0 (+-) 4i, and y(c) = Cos(4t) + Sin(4t)

using that fact I did what was necessary to enter that into the system of equations and now I can't figure out what I need to do in order to remove U1' and U2' from the equation separately in order to figure out what U2' and U1' are.

Any suggestions of what I should do next? What to multiply/divide/subtract/add/etc.


Should I have split up the Cos(4t) and Sin(4t) so that y1= Cos(4t), and y2 = Sin(4t)?

-WWB

Originally Posted by Whitewolfblue

I have a test coming up involving variation of parameters and I am practicing. I got stuck on this one, I know it should just be algebra but now I'm lost.

Using the system of equations:

U1'(y1)+U2'(y2) = 0
U1'(y1')+U2'(y2') = f(x)

I need to solve for U1' and U2'.

I started with y" + 16y = 4Tan(4t)

I found that r = 0 (+-) 4i, and y(c) = Cos(4t) + Sin(4t)

using that fact I did what was necessary to enter that into the system of equations and now I can't figure out what I need to do in order to remove U1' and U2' from the equation separately in order to figure out what U2' and U1' are.

Any suggestions of what I should do next? What to multiply/divide/subtract/add/etc.


Should I have split up the Cos(4t) and Sin(4t) so that y1= Cos(4t), and y2 = Sin(4t)?

-WWB

you should be removing the u primes from the equation and setting them equal to zero, and taking the derivative of whatever is left.

Afterwards you use the u primes as your system of equations, a bunch of stuff should cancel, if not you did something wrong.

you should be able to solve for all the primes individually as a system of equations or using the wronskian, although depending the wronskian can be much simpler.

What about this question as well? I'd rather not use wrongskian. Our instructor specifically requested we learn how to use VOP fully.

"Should I have split up the Cos(4t) and Sin(4t) so that y1= Cos(4t), and y2 = Sin(4t)?"

I think what I did was put the Y(c) into both y1 and y2.

-WWB

Originally Posted by Whitewolfblue

What about this question as well? I'd rather not use wrongskian. Our instructor specifically requested we learn how to use VOP fully.

"Should I have split up the Cos(4t) and Sin(4t) so that y1= Cos(4t), and y2 = Sin(4t)?"

I think what I did was put the Y(c) into both y1 and y2.

-WWB

no you leave the homogenous solution as a whole. You just differentiate the whole thing set all your u primes to zero and solve. Whatever is left is part of the particular.

Theres nothing really fully about it, and using the wrongskian is merely a place holder which saves time instead of solving it as a system of equations which ends up being the same thing because its a matrix.

you're y(c) is your homogenous, just differentiate and solve that using your system.

unless im misunderstanding your original question...

Originally Posted by p00ndawg

no you leave the homogenous solution as a whole. You just differentiate the whole thing set all your u primes to zero and solve. Whatever is left is part of the particular.

Theres nothing really fully about it, and using the wrongskian is merely a place holder which saves time instead of solving it as a system of equations which ends up being the same thing because its a matrix.

you're y(c) is your homogenous, just differentiate and solve that using your system.

unless im misunderstanding your original question...

I see what your talking about now.

Thanks,

-WWB