- #1

arhzz

- 251

- 49

- Homework Statement
- Solve the differential equation

- Relevant Equations
- -

Hello ! I need to solve this diffrential equation.

$$ y^{(4)} + 2y'' + y = 0 $$

First I wanted to find the homogenous solution,so I built the characteristic polynomial ( not sure if u say it so in english as well).I did that like this

$$\lambda^4 +2\lambda^2+1 = 0 $$. The solutins should be $$ \lambda_1 = i , \lambda_2 = -i $$ Now I used the approach to convert these complex solutions into sin and cos and I got this to be my homogenous solution.

$$y_h = c_1cos(x) + c_2sin(x) $$ Now this is not right,accorindg to wolfram alpha the solution should be

$$ y_h = (c_1+c_2x)cosx + (c_3+c_4x)sinx $$ Now the first thing that came to mind is multiplicty (not sure if it is the right word in english) but basically when the same solution appears more than once.But than I found this differential equation that I solved earlier today

$$ y'' +2y = 0 $$ Here the solution was +2i and -2i,and yet the homogenous solution had only 2 constants,

$$ y_h = c_1cos(2x)+c_2sin(2x) $$ and according to wolframalpha this is correct.I am obviously missing something but I am not sure what.

Unrelated note: The "preview" button is not working lately for me and it is incredibly annoying,whenever I click on it it does not show me a preview so I am writing "blindly" into LaTeX.Also when I reload the page something it buggs out not letting me write further,and latex literally not wanting to delete itself leading to very akward looking posts.Has any script been update,that I need to update on my browser? Because this started happening recently and its making life really annoying.

Thanks!

$$ y^{(4)} + 2y'' + y = 0 $$

First I wanted to find the homogenous solution,so I built the characteristic polynomial ( not sure if u say it so in english as well).I did that like this

$$\lambda^4 +2\lambda^2+1 = 0 $$. The solutins should be $$ \lambda_1 = i , \lambda_2 = -i $$ Now I used the approach to convert these complex solutions into sin and cos and I got this to be my homogenous solution.

$$y_h = c_1cos(x) + c_2sin(x) $$ Now this is not right,accorindg to wolfram alpha the solution should be

$$ y_h = (c_1+c_2x)cosx + (c_3+c_4x)sinx $$ Now the first thing that came to mind is multiplicty (not sure if it is the right word in english) but basically when the same solution appears more than once.But than I found this differential equation that I solved earlier today

$$ y'' +2y = 0 $$ Here the solution was +2i and -2i,and yet the homogenous solution had only 2 constants,

$$ y_h = c_1cos(2x)+c_2sin(2x) $$ and according to wolframalpha this is correct.I am obviously missing something but I am not sure what.

Unrelated note: The "preview" button is not working lately for me and it is incredibly annoying,whenever I click on it it does not show me a preview so I am writing "blindly" into LaTeX.Also when I reload the page something it buggs out not letting me write further,and latex literally not wanting to delete itself leading to very akward looking posts.Has any script been update,that I need to update on my browser? Because this started happening recently and its making life really annoying.

Thanks!