Method of Variation of Coefficients

[Husaaved]

Hello,

I am in an introductory undergraduate course on ODEs, currently on the method of variation of constants to solve nonhomogenous equations.

I am noticing that with many of these problems, when solving for constants after plugging in my guessed values for y I end up with enormous expressions, e.g. solving y'' - 4y' + 5y = 2(cos x)e^2x I ended up with the expression:

A(cos x(2e^2x + 4xe^2x + 2e^2x) - sinx(2xe^2x + e^2x) - sin x(2xe^2x + e^2x) + cos(xe^2x)) + B(sin x(4xe^2x + 2e^2x + 2e^2x) + cos x(2xe^2x + e^2x) + cos x(2xe^2x + e^2x) - sin x(xe^2x) - A(cos x(8xe^2x + 4e^2x) - sin x(4xe^2x) - B(sin x(8xe^2x + 4e^2x) + cos x(4xe^2x) - B(sin x(8xe62x + 4e^2x) + cosx(4xe^2x) + 5(Axe^2x)cos x + 5B(xe^2x)sin x = 2(cos x)e^2xI am not asking for my work to be checked as I ultimately arrived at the correct answer (y = e^2x(c1cos x + c2sin x) + (sinx)xe^2x), but is it entirely normal to obtain expressions this large? Are there methods learned later or in more advanced courses which streamline this process, or is this simply the nature of ODEs (i.e. since we are solving for functions we will inevitably end up with huge expressions)?

I'm only asking to make sure I am not missing something.

Thank you :)

 

[Simon Bridge]

...is it entirely normal to obtain expressions this large?

Depends on the field - but it is pretty common yes.

Are there methods learned later or in more advanced courses which streamline this process, or is this simply the nature of ODEs (i.e. since we are solving for functions we will inevitably end up with huge expressions)?

The advanced courses will deal more with problems that do not have such simple expressions. in general you can have f(y'',y',y,x)=0 ... and that f can be anything; so they can get arbitrarily complicated.

But you d get to notice things from all this experience.
For instance, in your example, the RHS looks like it is one of the solutions to the inhomogeneous equation.

 

[HallsofIvy]

I was a bit puzzled by your title "variation of coefficient"! That combines parts of two different methods of solving non-homogenous differential equations, "undetermined coefficients" and "variation of parameters".

"Undetermined coefficients" can by used to solve for a "particular solution" to, say, a linear differential equation with constant coefficients as long as the non-homogenous part is one of the functions that we expect as a solution to such a homogenous equation- polynomials, exponentials, sine or cosine, and products of those. If it is not- for example if the function is "tan(x)"- then that method will not work but we can use "variation of parameters".

 

[Husaaved]

Hello,

I realize this reply is very late -- sorry! I called it "varition of constants" because that is what it is called in one of the books I am using, "Multivariable Calculus, Linear Algebra, and Differential Equations" second edition by Stanley L. Grossman:

11.11 Nonhomogenous Equations: Variation of Constants*

That's the name of the section for that method, and the asterisk leads to a footnote which says "This procedure is also called the variation of parameters method, or Lagrange's method".

 

[Simon Bridge]

I called it "variation of constants" because...

Check the title to the thread: you used two different names ... but no matter.
It wasn't really that hard to figure out - just a caution to watch out for: small slips in a technical name can confuse or change the meaning.
If someone asks after what you mean, it's actually just a neutral request for information.

the asterisk leads to a footnote which says "This procedure is also called the variation of parameters method, or Lagrange's method".

FYI: the names in the footnote are more usual internationally... more modern authors would put the terms the other way around. i.e.

In mathematics, variation of parameters, also known as variation of constants, is a general method to solve inhomogeneous linear ordinary differential equations.
http://en.wikipedia.org/wiki/Variation_of_parameters​

Authors used to be able to get away with this more in, say, the mid 80's (when your book was published), but these days everyone is much more international. The text is old (pre-internet even) but cheap and only minor things have changed (mostly in style) since then - making it actually quite useful. I was thinking of basing a course off the 3rd edition. How are you liking the approach?

Just be alert about some things maybe being maybe slightly out of date.

 

[Husaaved]

Ahhh, I hadn't noticed. Thank you -- I'll know to be more mindful with that.

I actually really love the text! I'm taking all three classes as separate courses, and for multivariable we are using Stewart, for ODEs we're using An Introduction to Ordinary Differential Equations by Earl Coddington, and Lay for Linear Algebra, but I am mainly working through this text and the one by Coddington for all three classes. I like it because it connects the three subjects, and treats calculus 3 with a bit more rigor than Stewart. I have to supplement the linear algebra section with Schaum's though because it doesn't cover quite a few topics, but otherwise I was happy I came across it.

 

[Simon Bridge]

Nice to know - it's tough to judge a text from the other side - as it were.
Grossman is "proof light" - which is what I was mainly interested in, and the 3rd ed is on amazon for $10 or so so students can actually afford it. My tendency is to write my own texts - but that's a lot of writing for a subject so well covered elsewhere.
Cheers.