OpenStudy (anonymous):

need help using variation of parameters to solve simple D.E. y'+y=8e^(2x) need help using variation of parameters to solve simple D.E. y'+y=8e^(2x) @Mathematics

6 years ago
OpenStudy (jamesj):

So by the variation of parameters method, we want to try and find a particular solution that is a linear combination c1.y1 + c2.y2 + .... + cn.yn where the cj are constants and the yn are functions that are the derivatives of terms on the right hand side.

6 years ago
OpenStudy (jamesj):

In this case, there only one term on the RHS, e^(2x), and what's more its derivative is the same up to a constant.

6 years ago
OpenStudy (anonymous):

okay so do you need to use the roots to find 1 equation first?

6 years ago
OpenStudy (jamesj):

Hence the only term will need to try in the variation of parameters method is e^(2x); namely, try and find a solution of the form y = Ae^(2x)

6 years ago
OpenStudy (jamesj):

There's the homogeneous solution as well; we'll come back to that.

6 years ago
OpenStudy (jamesj):

That is, the solution of y' + y = 0; that is the homogeneous solution, call it yh (or sometimes it's called the complimentary solution yc).

6 years ago
OpenStudy (jamesj):

Once we've found the particular solution yp, the general solution to the equation is the sum y = c.yh + yp

6 years ago
OpenStudy (jamesj):

yh is easy with y' + y = 0. You've probably found it already; it's yh = e^(-x)

6 years ago
OpenStudy (jamesj):

Now coming back to the particular solution ... plug in our general form of the answer y = Ae^(2x) into the inhomogeneous equation and derive an equation in A. You do that step; I'll wait a minute for you.

6 years ago
OpenStudy (jamesj):

still there?

6 years ago
OpenStudy (anonymous):

sorry I had to switch computers.... so you use the general answers Ae^(2x) take the derivative and plug it back into the equation and set it equal to 8e^(2x)?

6 years ago
OpenStudy (anonymous):

i meant equation not answer

6 years ago
OpenStudy (jamesj):

yes. With y = Ae^(2x), substitute this into the equation y'+y=8e^(2x) and solve for A

6 years ago
OpenStudy (anonymous):

isn't that the same steps as the method of undermined coefficients? or do I just have them backwards

6 years ago
OpenStudy (jamesj):

In a way undetermined coefficients is a special case of variation of parameters; I guess I've been advising you here to use that special case, especially as the equation is linear and first order.

6 years ago
OpenStudy (jamesj):

Can you give me an example of using variation of parameters with a first order equation?

6 years ago
OpenStudy (anonymous):

I need to solve the same equation using both methods but when I tried I basically use the same steps.

6 years ago
OpenStudy (jamesj):

Are you suggesting we write y = u(x).e^(-x) and then solve for u?

6 years ago
OpenStudy (anonymous):

I guess but it seems pointless since they are the same thing

6 years ago
OpenStudy (jamesj):

We can do that if you want.

6 years ago
OpenStudy (jamesj):

More or less; but let's just do it. It's actually not a bad exercise.

6 years ago
OpenStudy (anonymous):

I just don't see what the difference is between the two approaches for this case

6 years ago
OpenStudy (jamesj):

If y = u(x)e^-x, then y' = u' e^-x - u e^-x Hence y' + y = 8e^(2x) implies u' e^-x - u.e^-x + u.e^-x = 8e^2x => u' e^-x = 8e^2x => u' = 8e^3x => u = (8/3)e^3x + c => y = (8/3)e^2x + c.e^-x And this indeed is the general solution.

6 years ago
OpenStudy (jamesj):

A third way to do it is with an integrating factor. That method would most closely imitate what I just wrote above.

6 years ago
OpenStudy (anonymous):

Ohhhh okay i think I see the difference now

6 years ago
OpenStudy (anonymous):

Thanks for your help!

6 years ago
OpenStudy (jamesj):

Sure. I actually had never thought of using VOP with first order equations before, so I was interested to see how this worked out.

6 years ago
OpenStudy (jamesj):

Pedagogically, this is not a bad exercise your professor is giving you.

6 years ago
OpenStudy (anonymous):

yeah it was the simplest problem on the hw and it gave me the most trouble.

6 years ago
OpenStudy (jamesj):

ok. Well, good luck with the rest.

6 years ago