Differential Equation and annihilator method

[Saladsamurai]

Homework Statement

Find the general solution for

Y''+16y=sec4x

Homework Equations

I was thinking of using annihilator method to find Yp, but I do not know how to annihilate inverse trig functions, only sine and cosine.

Is there a better approach to this to find Yp? Finding the complementary is easy enough.

But, figuring out which method to use to find Yp is always the hardest part of the problem for me.


Thanks,

 

[quantumdude]

Have you learned the method of variation of parameters yet? If so, then you might try multiplying both sides by cos(4x) and using it. I don't know if it will work, but it's worth a shot.

 

[sutupidmath]

Well, actually i have no idea what the annihilator method for finding Yp means, buti f you know how to annihilate sine and cosine than you can go from here, because there are no inverse trig functions involved in your equation. sec 4x=\frac{1}{cos 4x}

 

[HallsofIvy]

The "annihiliation method" is a technical term for "undetermined parateters".

And knowing that sec 4x=\frac{1}{cos 4x} doesn't help here.

Saladsamurai, the "annihilation method" or "undetermined coefficients" only works if you can "guess" the correct form ahead of time and you can really only do that when the function of x on the right side is one of the kinds of functions we expect to get as a solution to "linear homogenous d.e. with constant coefficients": exponentials, sine or cosine, polynomials, and combinations of those. For anything else, tangent, secant, logarithm, etc. You need to use "variation of parameters".

 

[Saladsamurai]

Well, actually i have no idea what the annihilator method for finding Yp means, buti f you know how to annihilate sine and cosine than you can go from here, because there are no inverse trig functions involved in your equation. sec 4x=\frac{1}{cos 4x}

Right, I meant reciprocal functions. But those cannot be annihilated. And by Yp<---Particular solution.

The "annihiliation method" is a technical term for "undetermined parateters".

And knowing that sec 4x=\frac{1}{cos 4x} doesn't help here.

Saladsamurai, the "annihilation method" or "undetermined coefficients" only works if you can "guess" the correct form ahead of time and you can really only do that when the function of x on the right side is one of the kinds of functions we expect to get as a solution to "linear homogenous d.e. with constant coefficients": exponentials, sine or cosine, polynomials, and combinations of those. For anything else, tangent, secant, logarithm, etc. You need to use "variation of parameters".

Yes. This method worked out well.

 

[Saladsamurai]

Right, I meant reciprocal functions. But those cannot be annihilated. And by Yp<---Particular solution.



Yes. This method worked out well.

I am still having trouble figuring out when to use what methods. So far, Variation of parameters is the only one I can really tell when to use. IF the RHS is Non-anihilatible, use Var of parameters.

But as for the others, it seems to be guess and check. Separation if variables, for instance, is used when 'the variables are seperable'...but when is that? I ask...well...when they are! There doesn't seem to be any "dead giveaways" that a DE is separable or not.

I think I will attempt to make a brief table of the methods I have learned so far along with instances in which to use those methods and then post it here to be 'peer edited'. I am sure I can make great use of the insight I would get from the PF members.

 

[HallsofIvy]

The "types of functions" which you can get as solutions to a "linear differential equation with constant coefficients" are:
exponentials
sine and cosine
polynomials
combinations of those: i.e. (x²-3 x)(e<sup>5x)(cos(x)- sin(x)) could be the a solution to (very complicated!) linear equation with constant coefficients.

If the function in a non-homogeneous equation, not multiplied by the dependent variable or any of its derivatives, is one of those, there are fairly standard rules for "guessing" the form you need to use "undetermined coefficients". If it not of that form, you will need to use "variation of parameters".</sup>

 

[FrogPad]

My first thought was just to apply the Laplace transform to solve (rather blindly). Intuition said I couldn't so I looked up some information:

Quote:
"It is unfortunate, but the Laplace Transform of the other trigonometric functions tan, cot, csc, and sec do not exists as they all have singularities for finite t."

source:
http://books.google.com/books?id=09...NCvLolz&sig=UA6OYViXd-QDshM62BSjTL0GPGU&hl=en

 

[Saladsamurai]

The "types of functions" which you can get as solutions to a "linear differential equation with constant coefficients" are:
exponentials
sine and cosine
polynomials
combinations of those: i.e. (x²-3 x)(e<sup>5x)(cos(x)- sin(x)) could be the a solution to (very complicated!) linear equation with constant coefficients.

If the function in a non-homogeneous equation, not multiplied by the dependent variable or any of its derivatives, is one of those, there are fairly standard rules for "guessing" the form you need to use "undetermined coefficients". If it not of that form, you will need to use "variation of parameters".</sup>

<sup>Alrighty. I think I would like to clarify this a little more. I think I am with you. I should use the method of undetermined coefficients when the RHS of the DE is of some form that its solution's form could be guessed, like P(x), sine, cosine, e^x and or any linear combination of those.

If the RHS is non-annihilable like, ln,tan, sec,csc I should use Variation of parameters.



What is it about the former that makes their solutions' forms "guessable" and the latter's that make them "unguessable?"</sup>

 

[HallsofIvy]

Because the are general rules you can use: For an exponential, e^rx, try that exponential, Ae^rx. For a polynomial of degree n, try all powers of x from n on down, Axⁿ+ Bx^n-1+ Cx^n-2+ ... + Yx+ Z. For sine or cosine or both, try Asin(rx)+ Bcos(rx). For combinations of those, try combinations. If one of those is already a solution of the associated homogeneous equation, multiply by t (if a double root, by t², etc.)

I've alway thought of the Laplace transform as a very "mechanical" way to get solutions you could get by more fundamental methods anyway. Suited for Engineers perhaps but not mathematicians.

 

[FrogPad]

Because the are general rules you can use: For an exponential, e^rx, try that exponential, Ae^rx. For a polynomial of degree n, try all powers of x from n on down, Axⁿ+ Bx^n-1+ Cx^n-2+ ... + Yx+ Z. For sine or cosine or both, try Asin(rx)+ Bcos(rx). For combinations of those, try combinations. If one of those is already a solution of the associated homogeneous equation, multiply by t (if a double root, by t², etc.)

This sounds mechanical to me too.


I'd imagine all methods have a place. My preference is to use whatever is faster. I will say though, that after taking a controls class the Laplace transform is extremely useful for gaining insight into a differential equation.

But yeah, it's engineering stuff... not math stuff. Math stuff is too hard! hehe