Please can someone solve my differential equation with undetermined coefficients

[mariofk]

Hi,this is my first post.I really need to solve this problem because my professor insists on it.He said that I solve it using undetermined coefficients(like C1 and C2).I tried a lot of times and I always couldnt find a way to solve it completely.I would really appreciate if someone could help me.
This is the task

Cheers.

 

[Deleted member 4993]

Hi,this is my first post.I really need to solve this problem because my professor insists on it.He said that I solve it using undetermined coefficients(like C1 and C2).I tried a lot of times and I always couldnt find a way to solve it completely.I would really appreciate if someone could help me.
This is the task
View attachment 18776
Cheers.

Please show us what you have tried and exactly where you are stuck.

Please follow the rules of posting in this forum, as enunciated at:

https://www.freemathhelp.com/forum/threads/read-before-posting.109846/#post-486520

Please share your work/thoughts about this assignment.

 

[HallsofIvy]

IF you really have "tried a lot of times" you should have included at least one of that attempts. That would let us know what you do understand about this problem, where you need help- at least that you know what "undetermined coefficients" means!

First, the "associated homogeneous equation" is y''+ 2y'- 3y= 0. That has characteristic equation \(\displaystyle r^2+ 2r- 3= (r- 1)(r+ 3)= 0\).

So what is the general solution to that associated homoteneous equation?

Now all you need to do is find a single solution to the entire equation and add it to that general solution. Since the right hand side is sin(x) and I know that the derivative of sin(x) is cos(x) and the derivative of cos(x), I would try a solution of the form y= Acos(x)+ Bsin(x) (A and B are the "undetermined coefficients). What do you get when you take the first and second derivatives of that and put them into the equation?