Why is my particular solution not matching with the book's answer?

[newtomath]

Given t^2 y'' -t(t+2)y' = (t+2)y= 2t^3 and y1= t, y2= te^t
Find the particular solution-

I ve worked the problem to [ -2t^2 -2t] by:
-t * Integral [ 2t* te^t/ t^2e^t] + te^t * Integral [ 2t^2/ t^2e^t]


whereas the book states that it is simply -2t^2. Can you guys tell me where I made my mistake, I am stumped.

 

[timthereaper]

Given t^2 y'' -t(t+2)y' = (t+2)y= 2t^3 and y1= t, y2= te^t

Is that first = supposed to be +?

 

[pmsrw3]

Given t^2 y'' -t(t+2)y' = (t+2)y= 2t^3 and y1= t, y2= te^t
Find the particular solution-

I ve worked the problem to [ -2t^2 -2t] by:
-t * Integral [ 2t* te^t/ t^2e^t] + te^t * Integral [ 2t^2/ t^2e^t]

whereas the book states that it is simply -2t^2. Can you guys tell me where I made my mistake, I am stumped.

To be honest, I don't completely understand this, but I'm not sure you've actually done anything wrong. You got the particular solution -2t^2-2t, and the book says -2t^2. But notice that one of the homogeneous solutions is t. If you add a linear combination of the homogeneous solutions to the particular solution, you get another equally valid particular solution. So, if -2t^2 -2t works, so does -2t^2 -2t + 2t = -2t^2.

 

[MathematicalPhysicist]

Well a private solution here will be of the form:
At^3+Bt^2+Ct
where you can see quite immediately that A=0 (cause it has a term of t^4 which obviously gets nullified).

I did the calculation and got exactly as the book, try again.
PS
the C term doesn't get canceled though.

 

[blue_raver22]

There could be a few reasons why your solution is not matching with the book's answer. First, it is possible that you made a mistake in your calculations. Double check your work and make sure you haven't made any errors.

Another possibility is that the book's solution is simplified or manipulated in a way that is not immediately obvious. It is important to understand the steps and methods used in solving the problem, rather than just focusing on the final answer.

It is also possible that there is more than one way to solve the problem, and both your solution and the book's solution are correct but just written differently.

Lastly, it is important to note that in mathematics and science, there can be more than one correct answer to a problem. As long as your solution follows the rules and principles of the subject, it can be considered a valid solution.

Overall, it is important to carefully review your work and try to understand the steps and methods used in solving the problem, rather than just focusing on the final answer. Don't be discouraged if your solution does not match with the book's answer, as long as you can justify your approach and it follows the rules and principles of the subject.