# Variation of Parameters to solve a second order ODE

• Bonnie
In summary, the person is trying to solve for u1 and u2 in terms of y1 and y2, but they keep getting complicated expressions involving e's and i's. They need to use the trigonometric form of the equation and substitute in y1 and y2.
Bonnie

## Homework Statement

The question I am working on is the one in the file attached.

## Homework Equations

y = u1y1 + u2y2 :

u1'y1 + u2'y2 = 0
u1'y1' + u2'y2' = g(t)

## The Attempt at a Solution

I think I have got part (i) completed, with y1 = e3it and y2 = e-3it. This gives a general solution to the homogeneous equation to be y = C1cos3t + C2sin3t.
For part (ii) I know that for variation of parameters you need to substitute y1 and y2 and their derivatives into the above system of equations to solve for u1' and u2', then integrate these to find u1 and u2, from which you can get the desired solution of
y = yp + yh
But I find that when I try to do that by substituting y1 = e3it and y2 = e-3it I just end up with very complicated expressions involving lots of e's and i's which the desired solution does not contain. I know it involves combining the information about the general soln from (i) with the method I have described, but I just don't know how to make it work. Any help would be appreciated, thank you in advance!

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I just end up with very complicated expressions involving lots of e's and i's which the desired solution does not contain
So the thing to do is avoid these exponential ##y## and take the trigonometric ##y## that you see in front of you when you write ##y = C_1\cos 3t + C_2\sin3t\ ## !

http://tutorial.math.lamar.edu/Classes/DE/VariationofParameters.aspx

Bonnie said:

## Homework Statement

The question I am working on is the one in the file attached.

## Homework Equations

y = u1y1 + u2y2 :

u1'y1 + u2'y2 = 0
u1'y1' + u2'y2' = g(t)

## The Attempt at a Solution

I think I have got part (i) completed, with y1 = e3it and y2 = e-3it. This gives a general solution to the homogeneous equation to be y = C1cos3t + C2sin3t.
For part (ii) I know that for variation of parameters you need to substitute y1 and y2 and their derivatives into the above system of equations to solve for u1' and u2', then integrate these to find u1 and u2, from which you can get the desired solution of
y = yp + yh
But I find that when I try to do that by substituting y1 = e3it and y2 = e-3it I just end up with very complicated expressions involving lots of e's and i's which the desired solution does not contain. I know it involves combining the information about the general soln from (i) with the method I have described, but I just don't know how to make it work. Any help would be appreciated, thank you in advance!
Have you forgotten that ##e^{\pm i \theta} = \cos \theta \pm i \sin \theta \,?##

Ray Vickson said:
Have you forgotten that ##e^{\pm i \theta} = \cos \theta \pm i \sin \theta \,?##
Yes I had, thank you. I'll have another try!

## 1. What is the Variation of Parameters method for solving second order ODEs?

The Variation of Parameters method is a technique used to find a particular solution to a second order ordinary differential equation (ODE). It involves finding a new set of functions that satisfy the ODE and using them to construct the particular solution.

## 2. When should the Variation of Parameters method be used instead of other techniques?

The Variation of Parameters method is most useful when the coefficients of the ODE are not constant and cannot be easily separated. It is also helpful when the homogeneous solution is already known.

## 3. How does the Variation of Parameters method work?

The method involves finding two linearly independent functions, typically denoted as u1(x) and u2(x), that satisfy the homogeneous equation. Then, a particular solution y_p(x) is constructed using these functions and a set of coefficients that are determined through substitution into the original equation.

## 4. Can the Variation of Parameters method be used for higher order ODEs?

Yes, the Variation of Parameters method can be extended to solve higher order ODEs. However, the process becomes more complex as the number of equations and unknown coefficients increases.

## 5. Are there any limitations to the Variation of Parameters method?

One limitation is that the method can only be used for linear ODEs. Additionally, it may not always be possible to find a set of functions that satisfy the ODE, in which case another method must be used to find the particular solution.

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