Why Are u and v Functions Zero in VOP for Homogeneous 2nd Order Equations?

Click For Summary
SUMMARY

The discussion focuses on the application of Variation of Parameters (VOP) to solve homogeneous second-order differential equations. It highlights that when using VOP, the functions u and v are zero due to the nature of the coefficients being constant and the differential equation equating to zero. The uniqueness theorem confirms that VOP yields a valid and unique solution for these equations, despite the method being more time-consuming than others.

PREREQUISITES
  • Understanding of homogeneous second-order differential equations
  • Familiarity with the Variation of Parameters (VOP) method
  • Knowledge of the uniqueness theorem in differential equations
  • Basic concepts of differential equation coefficients
NEXT STEPS
  • Study the application of Variation of Parameters in non-homogeneous equations
  • Explore the uniqueness theorem in greater detail
  • Learn about alternative methods for solving second-order differential equations
  • Investigate the implications of constant coefficients in differential equations
USEFUL FOR

Mathematics students, educators, and professionals dealing with differential equations, particularly those focusing on methods for solving homogeneous second-order equations.

lonewolf219
Messages
186
Reaction score
2
I just realized you can use variation of parameters (VOP) to solve for homogeneous 2nd order equations. I see it takes much longer to do so. But I was wondering why, if you use VOP, the u and v functions are 0. Is this because the coefficients of the homogeneous equation are constant, or possibly because the differential equation is equal to 0 to begin with?
 
Physics news on Phys.org
One way to look at it is simply to note that VOP results in a valid solution to the problem. The uniqueness theorem assures you that you have the only correct solution.
 

Similar threads

Undergrad 2nd order ODE's, variation of parameters and the notorious constraint
  • · Replies 5 ·
Replies
5
Views
2K
Graduate Variation of Parameters for System of 1st order ODE
  • · Replies 1 ·
Replies
1
Views
2K
Undergrad On vibrating string and differentiating infinite sum
  • · Replies 7 ·
Replies
7
Views
3K
Undergrad Explanation of all "its linearly independent derivatives"
  • · Replies 3 ·
Replies
3
Views
3K
Undergrad Number of solutions to the order of the equation
  • · Replies 2 ·
Replies
2
Views
3K
Undergrad How to do Magnus expansion for time-dependent companion matrix?
  • · Replies 7 ·
Replies
7
Views
3K
MHB Method of Reduction of Order and Variation of Parameters
  • · Replies 2 ·
Replies
2
Views
5K
Undergrad Second order non-homogeneous linear ordinary differential equation
  • · Replies 2 ·
Replies
2
Views
3K
Undergrad Why Lagrange’s method of solving Pp + Qq=R works?
  • · Replies 3 ·
Replies
3
Views
2K
Undergrad Proving a basis is a basis for homogeneous linear differential equation with constant (complex) coefficients
  • · Replies 2 ·
Replies
2
Views
2K