2nd-order nonhomogeneous equation

[yecko]

Homework Statement

Homework Equations

2nd order nonhomogeneous equation

The Attempt at a Solution

(answers typed in in the pic)
for part c, if I can't put Ate^(2t) & Be^(2t), how is the form of particular equation like?
I have used Y=(At+B)e^(2t)+(Ct+D), because of the largest power of the equation.
Thank you.

 

[scottdave]

The derivative of Be^(2t) should be 2Be^(2t),
Then you should have this for y':
Ae^(2t) + 2Ate^(2t) + 2Be^(2t) + C

Then your y'' will change as it is the derivative of y'.

 

[Ray Vickson]

Homework Statement

View attachment 221738

Homework Equations

2nd order nonhomogeneous equation

The Attempt at a Solution

(answers typed in in the pic)
for part c, if I can't put Ate^(2t) & Be^(2t), how is the form of particular equation like?
I have used Y=(At+B)e^(2t)+(Ct+D), because of the largest power of the equation.
Thank you.

Do NOT post pictures of problems/equations. Your images are unreadable on my screen, and I suspect that almost all helpers will be willing to assist you. Just type out the main parts of the problem---you don't need to type everything, just the most important parts. After all, if you want us to volunteer out time freely you can at least make the effort to present your problem clearly.

 

[yecko]

The derivative of Be^(2t) should be 2Be^(2t),
Then you should have this for y':
Ae^(2t) + 2Ate^(2t) + 2Be^(2t) + C

Then your y'' will change as it is the derivative of y'.

other than my fault in deriving Y', my Y answer is also wrong
for Y, the comment is mentioned that the answer is without Ate^(2t) or Be^(2t), how can a particular equation without this two variables? thanks

 

[yecko]

Do NOT post pictures of problems/equations. Your images are unreadable on my screen, and I suspect that almost all helpers will be willing to assist you. Just type out the main parts of the problem---you don't need to type everything, just the most important parts. After all, if you want us to volunteer out time freely you can at least make the effort to present your problem clearly.

question:

In this problem you will use undetermined coefficients to solve the nonhomogeneous equation y''-4y'+4y=18te^(2t)-2e^(2t)-(4t+4)
with initial values y(0)=-3 and y'(0)=-6
Write the form of the particular solution and its derivatives. (Use A, B, C, etc. for undetermined coefficients. Y=?, Y'=?, Y''=?

sorry for not typing out in advance, because the screenshot gave the questions, my attempts, and comments which I thought it would be more comprehensive.

 

[yecko]

other than my fault in deriving Y', my Y answer is also wrong
for Y, the comment is mentioned that the answer is without Ate^(2t) or Be^(2t), how can a particular equation without this two variables? thanks

ok, finally i made it A+Bt+Ce^(2t)t^2+De^(2t)t^3 which is correct... why and when should I multiply t/t^2 to the particular equation? thanks

 

[Mark44]

ok, finally i made it A+Bt+Ce^(2t)t^2+De^(2t)t^3 which is correct... why and when should I multiply t/t^2 to the particular equation? thanks

Take a look at this Insights article I wrote a while back -- https://www.physicsforums.com/insights/solving-nonhomogeneous-linear-odes-using-annihilators/

 

[Ray Vickson]

ok, finally i made it A+Bt+Ce^(2t)t^2+De^(2t)t^3 which is correct... why and when should I multiply t/t^2 to the particular equation? thanks

Because it works! There is no better reason than that.

A hundred years ago some smart people figured out that we should do such things, and all we need to do is take their advice. Basically, that is what education is all about.

You might well ask how these things were discovered in the first place. The answer, I am afraid, is that people try many different methods; some of them fail and are a waste of time, while others work. What you see in print are only the results of the successes, not the hours of wasted time and mountains of scrap paper that went into the exploration.