Assumptions about particular solutions

[leehufford]

Hello,

Working through some reduction of order problems. I'm not sure about why the structure of a particular solution is assumed. Here's what I mean:

Given y"-4y = 2 and a known solution is e^-2x, use reduction of order to find a second solution and a particular solution.

Using a formula, the second solution is found quite easily to be y= ce^2x. But the particular solution is assumed to be of the form y = Ax + B. Taking a few derivatives and substituting yields a particular solution of y = -1/2.

But why do we assume the particular solution is of the form y = Ax + B? Thanks in advance,

Lee

 

[Simon Bridge]

Because - historically, that's been found to be a good guess.

To work out what solution to guess, you use the form of the inhomogeniety as a guide.
People have been working on these sorts of problems for long enough for some patterns to emerge so you don't have to use trial and error every single time.

You seem to have a typo in your example.
I'm guessing you meant to write y''+4y=2

The inhomogeniety is a polynomial of degree N=0.
y_p=Ax+B would be a polynomial of degree 1=N+1

http://www.math.uah.edu/howell/DEtext/Part3/Guess_Method.pdf
if y is any polynomial of degree N , then y , y′ and y′′ are also polynomials of degree N or less.

Since we want this to match the right side of the above differential equation, which is a
polynomial of degree N , it seems reasonable to [guess] a polynomial of degree N...
... seems to disagree with the choice, which is guessing a polynomial of degree N+1.

 

[leehufford]

Thank you for the reply.

Yes that was a typo, and yes your assumption of what I meant to type was correct.

So are you saying when the nonhomogenous part is of degree N I should guess a particular solution with degree N? I pretty much only chose N+1 because that's what my teacher did.

But when equating coefficients the Ax term drops out leaving only B (-1/2). So the conclusion I drew from your response is to basically pick an N equal to the N of the nonhomogenous part and any higher N (like what I did) is unnessesary work?

Thanks again for the swift, accurate, quality response.

-Lee

 

[Simon Bridge]

So are you saying when the nonhomogenous part is of degree N I should guess a particular solution with degree N? I pretty much only chose N+1 because that's what my teacher did.

Does it matter ... what efect does it have on the result if you choose a polynomial of degree N or bigger?

Oh I see this already occurred to you and you checked it out - good initiative!

But when equating coefficients the Ax term drops out leaving only B (-1/2). So the conclusion I drew from your response is to basically pick an N equal to the N of the nonhomogenous part and any higher N (like what I did) is unnessesary work?

... that's the one :)
Well done.

 

[blue_raver22]

Hello Lee,

Thank you for reaching out with your question. It's great to see that you are working through reduction of order problems and seeking to understand the reasoning behind the assumptions made in finding a particular solution.

To answer your question, the reason why we assume a particular solution of the form y = Ax + B in this case is because it is a common form of solution for linear differential equations. In general, when we are solving a linear differential equation of the form y" + p(x)y' + q(x)y = f(x), we assume a particular solution of the form y = Ax + B if f(x) is a polynomial of degree n (where n is the order of the differential equation). This is known as the method of undetermined coefficients.

In your specific example, the given equation y"-4y = 2 is a second-order linear differential equation, and the right-hand side f(x) is a constant (which can be seen as a polynomial of degree 0). Therefore, we assume a particular solution of the form y = Ax + B. By substituting this particular solution into the equation and solving for the coefficients A and B, we can find the particular solution that satisfies the given equation.

I hope this explanation helps you understand the reasoning behind the assumption of a particular solution in this type of problem. If you have any further questions, please let me know. Keep up the good work in your studies!

Best,