Converting an nth order equation to a system of first order equations

In summary, the homework statement is trying to find a solution to a system of first order equations with initial conditions of y(0)=0 and y'(0)=7. The attempt at a solution is first done by isolating the highest derivative of y''=-x^2y'-12y. Next, the problem is converted to a system of first order equations with initial conditions of y(0)=0 and y'(0)=7. The system is then written in matrix form and solved. The result is [u1(0),u2(0)]=[y(0),y'(0)] which is [0,7].
  • #1
hachi_roku
61
0

Homework Statement


convert y'' +x^2y'+12y=0 to a system of first order equations with initial conditions y(0)=0 y'(0)=7.


Homework Equations





The Attempt at a Solution


first i isolate highest derivative y'' = -x^2y'-12y
then i let u_1=y u_2=y'

then (u_1)' = u_2 and (u_2)= y''

then (u_2)' = (-12u_1)-(x^2u_2)

i then write these as

(u_1)' = 0*u_1 + 1u_2
(u_2)' = (-12u_1) + (-x^2u_2)


so then in matrix form i have

matrix [u_1 u_2] = [top -0 1 bottom -12 -x^2] *[u_1 u_2] + [0 0]

i think I am close but i don't know how to get vector c but putting vec u(0) ...pls help and sorry for the poor notation...i can rewrite but i can't find link to use the symbols and such
 
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  • #2
bump...pleaseee help
 
  • #3
I think you've basically got it. [u1,u2]'=[[0,1],[-12,-x^2]]*[u1,u2]. You can add [0,0] to that but there's no need to. The initial condition is then [u1(0),u2(0)]=[0,7], right? If you want some texing clues check out https://www.physicsforums.com/showthread.php?t=8997
 
  • #4
im sorry but if u can, please help me with that matrix algebra...my prof just copied chicken scratch notes and didn't explain anything.

edit :this is what i get

[u1,u2]'= [top 0 u1 bottom -12u2 -x^2u2]

how to i get my final answer
 
Last edited:
  • #5
There's not much to explain. The problem was to convert the problem to a system of first order ode's. You have already done that, I think. It didn't say you should solve it, right? It just said convert.
 
  • #6
your right, it says to convert, but my prof also wants us to find vector c with the initial conditions (forgot to mention). that's how I am not sure you get [0,7] which is the correct answer
 
  • #7
[u1(0),u2(0)]=[y(0),y'(0)], that was your definition of u1 and u2, right?
 
  • #8
yes.
 
  • #9
hachi_roku said:
yes.

Ok, so I'm guessing you also see why that's [0,7].
 
  • #10
ok i think i get it... the y values from the initial conditions are the ones put in the vector c matrix
 
  • #11
got it thanks!
 

What is the purpose of converting an nth order equation to a system of first order equations?

The purpose of converting an nth order equation to a system of first order equations is to simplify the problem and make it easier to solve. This method breaks down a complex problem into smaller, more manageable parts, allowing for a step-by-step approach to finding the solution.

How is an nth order equation converted to a system of first order equations?

To convert an nth order equation to a system of first order equations, we introduce new variables and rewrite the equation as a set of differential equations. This involves breaking down the equation into its individual terms and expressing each term in terms of the new variables.

What are the benefits of using a system of first order equations?

Using a system of first order equations allows for a more systematic and organized approach to solving complex problems. It also makes it easier to apply numerical methods and computer algorithms to find solutions. Additionally, it can help to reveal relationships and patterns within the problem that may not have been apparent in its original form.

Are there any limitations to converting an nth order equation to a system of first order equations?

While converting an nth order equation to a system of first order equations can be a useful tool, it may not always be possible or practical. Some equations may not have a suitable form for conversion or may result in a system that is too complex to solve. In these cases, alternative methods may need to be used.

Can a system of first order equations be converted back to an nth order equation?

In most cases, it is possible to convert a system of first order equations back to an nth order equation. However, the process can be time-consuming and may not always result in a simplified form. It is generally more efficient to solve the problem using the system of first order equations rather than converting it back to its original form.

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