Form of a particular solution for N.H.L.D.E. w/ constant coefficients

In summary, the conversation discusses finding the form of a particular solution for a linear, differential equation with given roots of the auxiliary equation. The method of undetermined coefficients is used, but the coefficients are not calculated. The solution is verified as correct.
  • #1
jegues
1,097
3

Homework Statement



You are given that the roots of the auxiliary equation associated with the linear, differential equation

[tex]\phi(D)y = 2x- 3xe^{-3x}[/tex]

are [tex]m = \pm2,0,0.[/tex] Write down the form of a particular solution of the differential equation as predicted by the method of undetermined coefficients. Do NOT find the coefficients, just the form of the particular solution.

Homework Equations





The Attempt at a Solution



See figure attached for my attempt at the solution. I'm not entirely convinced I'm doing this properly so I'd just like for someone to verify my work.

Thanks again!
 

Attachments

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  • #2
Not sure what the various rules you refer to are, but the answer and reasoning look right to me.
 

Related to Form of a particular solution for N.H.L.D.E. w/ constant coefficients

1. What is a non-homogeneous linear differential equation (N.H.L.D.E.) with constant coefficients?

A non-homogeneous linear differential equation with constant coefficients is a type of differential equation that involves a function and its derivatives, where the coefficients of the derivatives are constants. It is called non-homogeneous because it includes a function that is not equal to zero, unlike a homogeneous differential equation where the function is equal to zero.

2. What is the form of a particular solution for N.H.L.D.E. with constant coefficients?

The form of a particular solution for N.H.L.D.E. with constant coefficients is a general solution that satisfies the given differential equation. It is usually in the form of a polynomial or a combination of exponential and trigonometric functions.

3. How do you find the particular solution for N.H.L.D.E. with constant coefficients?

To find the particular solution for N.H.L.D.E. with constant coefficients, you can use the method of undetermined coefficients or variation of parameters. The method of undetermined coefficients involves guessing a particular solution based on the form of the non-homogeneous term, while variation of parameters involves finding a particular solution by adding a multiple of the homogeneous solution to a particular integral.

4. Can the constant coefficients in N.H.L.D.E. be variable?

No, the constant coefficients in N.H.L.D.E. cannot be variable. The term "constant" means that the coefficients do not depend on the independent variable. Variable coefficients would make the differential equation non-linear, and the methods used for solving linear equations would not apply.

5. What are some real-world applications of N.H.L.D.E. with constant coefficients?

N.H.L.D.E. with constant coefficients are commonly used in physics, engineering, and other fields to model real-world situations. Examples include the motion of a pendulum, the growth of a population, and the decay of a radioactive substance. These equations help us understand and predict the behavior of systems over time.

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