# Variation of Parameter Problem

• bengaltiger14
In summary, the conversation discusses finding the general solution using the method of variation of parameters for a differential equation with repeated roots. The solution involves using two independent solutions to the associated homogeneous equation and following the usual procedure for variation of parameters.
bengaltiger14

## Homework Statement

Find the general solution using the method of variation of parameters of:

y''-6y'+9y=(x^-3)(e^3x)

I found the roots of the corresponding homogeneous equation to be lamba = 3. So there are repeated roots. My question is, how do I solve a variation of parameter question with repeated roots? I know how to do it using reduction of order but confused on variation of parameters

The same way you solve it without repeated roots! Since x= -3 is a double root of the characteristic equation, two independent solutions to the associated homogeneous equation are $e^{-3x}$ and $xe^{-3x}$.

Look for a solution to the entire equation of the form $y(x)= u(x)e^{-3x}+ v(x)xe^{-3x}$. Now just follow the usual procedure for variation of parameters.

## 1. What is the "Variation of Parameter Problem"?

The Variation of Parameter Problem (VPP) is a mathematical concept used in the field of differential equations. It involves finding a particular solution to a non-homogeneous differential equation by using a set of functions, called the "variation of parameters". It is also known as the "method of undetermined coefficients".

## 2. When is the "Variation of Parameter Problem" used?

The VPP is used when solving non-homogeneous linear differential equations, where the coefficients are constants. It is particularly useful when the non-homogeneous term is a function that can be expressed as a linear combination of known functions.

## 3. How is the "Variation of Parameter Problem" solved?

To solve the VPP, one must first find the homogeneous solution to the differential equation. Then, a set of functions, called the "variation of parameters", is used to form a particular solution. These functions are then substituted into the original differential equation, and their coefficients are determined by solving a system of linear equations.

## 4. What are the advantages of using the "Variation of Parameter Problem" method?

The VPP method allows for a systematic and straightforward approach to solving non-homogeneous differential equations. It also provides a general solution for the particular solution, rather than just a single solution. Additionally, it can be used for a wide range of non-homogeneous functions.

## 5. Are there any limitations to the "Variation of Parameter Problem"?

While the VPP method is useful for solving many non-homogeneous differential equations, it is not applicable to all types of differential equations. It also requires a certain level of mathematical knowledge and skill to solve, which may be a limitation for some users. Finally, it can be time-consuming and may not always provide a closed-form solution.

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