ODE: Combining Undetermined Coeff. & VOP Method

In summary, the conversation discusses combining two methods to solve an ODE and suggests using the VOP method and undetermined coefficients. It also mentions finding two particular solutions and adding them to the complementary solution. The conversation concludes with a thank you for the helpful insight.
  • #1
gabriels-horn
92
0
Title should read "Combining", is there anyway a moderator could alter that so the search function isn't messed up?

Homework Statement


Image4.jpg


The Attempt at a Solution


I am familiar with both methods, however combining the two is foreign to me. Anyone have any suggestions for this ODE? My guess would be to use the VOP method for the (1/x)*e^x portion and the undetermined coefficients for the 4x^2-3 portion. Any pointers?
 
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  • #2


You have the right idea. I will call the left side of your equation L(y) for brevity.

Your complementary solution yc satisfies the homogeneous equation

L(y) = 0. Now if you have two particular solutions satisfying L(yp1) = f(x) and L(yp2) = g(x), then

L(yp1+yp2) = L(yp1)+L(yp1) = f(x)+g(x)

so find the two particular solutions separately as you have indicated, and add them to your yc.
 
  • #3


LCKurtz said:
You have the right idea. I will call the left side of your equation L(y) for brevity.

Your complementary solution yc satisfies the homogeneous equation

L(y) = 0. Now if you have two particular solutions satisfying L(yp1) = f(x) and L(yp2) = g(x), then

L(yp1+yp2) = L(yp1)+L(yp1) = f(x)+g(x)

so find the two particular solutions separately as you have indicated, and add them to your yc.

Great, thanks for the insight.
 

1. What is the purpose of using the method of undetermined coefficients in solving ODEs?

The method of undetermined coefficients is used to solve non-homogeneous linear differential equations by finding a particular solution that satisfies the non-homogeneous terms of the equation. This method is useful for finding solutions to a wide range of ODEs, including those with polynomial, trigonometric, and exponential functions.

2. How does the method of undetermined coefficients work?

The method of undetermined coefficients involves making an educated guess for the form of the particular solution based on the non-homogeneous terms in the ODE. This guess is then substituted into the original equation, and the coefficients are determined by comparing like terms. The particular solution is then added to the complementary solution to obtain the general solution.

3. What is the VOP method and how is it used in solving ODEs?

The VOP (Variation of Parameters) method is used to find a particular solution for non-homogeneous linear differential equations. This method involves finding a set of functions, known as variation of parameters, that are substituted into the complementary solution to obtain the particular solution. These parameters are then solved for by using initial conditions or boundary conditions.

4. Can the method of undetermined coefficients and VOP method be combined?

Yes, the method of undetermined coefficients and VOP method can be combined to solve more complex ODEs. This combination is useful in cases where the non-homogeneous terms are a combination of functions that cannot be easily solved using either method alone.

5. Are there any limitations to using the method of undetermined coefficients and VOP method?

While the method of undetermined coefficients and VOP method are powerful techniques for solving ODEs, they do have some limitations. These methods are only applicable to linear differential equations and cannot be used for non-linear equations. Additionally, they may not be able to find solutions for all types of non-homogeneous terms, such as those with repeated roots or non-constant coefficients.

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