2nd-order nonhomogeneous equation

In summary, people tried many different methods to solve this particular nonhomogeneous equation and found one that worked.
  • #1
yecko
Gold Member
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15

Homework Statement


螢幕快照 2018-03-10 上午11.05.39.png


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.
 

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  • #2
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'.
 
  • #3
yecko said:

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.
 
  • #4
scottdave said:
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
 
  • #5
Ray Vickson said:
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.
 
  • #6
yecko said:
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
 
  • #8
yecko said:
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.
 
Last edited:

Related to 2nd-order nonhomogeneous equation

What is a 2nd-order nonhomogeneous equation?

A 2nd-order nonhomogeneous equation is a type of differential equation that involves a second derivative of a variable and a non-zero constant term. It is considered nonhomogeneous because the equation is not equal to zero, unlike a homogeneous equation.

How is a 2nd-order nonhomogeneous equation solved?

There are various methods for solving a 2nd-order nonhomogeneous equation, including the method of undetermined coefficients, variation of parameters, and Laplace transform. The specific method used depends on the form of the equation and the given initial conditions.

What is the difference between a homogeneous and nonhomogeneous equation?

A homogeneous equation is one where the right-hand side is equal to zero, while a nonhomogeneous equation has a non-zero constant term on the right-hand side. This difference affects the methods used to solve the equation.

Why are 2nd-order nonhomogeneous equations important in science?

2nd-order nonhomogeneous equations are important in science because they are often used to model physical systems and phenomena. By solving these equations, scientists can gain insight into the behavior and properties of these systems, which can then be applied to real-world situations.

What are some examples of 2nd-order nonhomogeneous equations in science?

Some examples of 2nd-order nonhomogeneous equations in science include the equations used to model oscillations in a mass-spring system, the motion of a pendulum, and the growth of a population over time. These equations can also be found in fields such as engineering, economics, and biology.

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