Variation of Parameter Problem

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.
  • #1
bengaltiger14
138
0

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
 
Physics news on Phys.org
  • #2
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 [itex]e^{-3x}[/itex] and [itex]xe^{-3x}[/itex].

Look for a solution to the entire equation of the form [itex]y(x)= u(x)e^{-3x}+ v(x)xe^{-3x}[/itex]. 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.

Similar threads

  • Calculus and Beyond Homework Help
Replies
7
Views
486
  • Calculus and Beyond Homework Help
Replies
2
Views
162
  • Calculus and Beyond Homework Help
Replies
3
Views
1K
  • Calculus and Beyond Homework Help
Replies
13
Views
244
  • Calculus and Beyond Homework Help
Replies
2
Views
860
  • Calculus and Beyond Homework Help
Replies
1
Views
1K
  • Calculus and Beyond Homework Help
Replies
3
Views
1K
  • Calculus and Beyond Homework Help
Replies
3
Views
2K
  • Calculus and Beyond Homework Help
Replies
1
Views
2K
  • Calculus and Beyond Homework Help
Replies
6
Views
2K
Back
Top