Method of undetermined coefficients

In summary, the particular solution for the given ODE using the method of undetermined coefficients is y(t) = -2sin(t) + 1. The homogeneous solution can be found by solving the characteristic equation and is given by y = c1*exp(-1/4*t)cos(sqrt(15
  • #1
cloud18
8
0
y" + 0.5y' + y = 1-cos(t); y(0) = y'(0) = 0

I used method of undetermined coefficients to get particular solution:

Y(t) = -2sin(t) + 1

To get homogeneous solution, I solved characteristic equation to get complex roots:

r_1,2 = -1/4 +- i*sqrt(15)/4

so homogeneous solution is:

y = c1*exp(-1/4*t)cos(sqrt(15)/4*t) + c2*exp(-1/4*t)sin(sqrt(15)/4*t)

But when I use the initial conditions, I get c1 = c2 = 0.
Is this right, or did I make a mistake?

I simplified the original problem with an easier forcing function to try first, so it is possible the ODE doesn't
make sense?
 
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  • #2
Yes, you made a mistake! You didn't add the particular solution to the solution to the associated homogeneous equation!
The general solution to the entire equation is y(t)= c1*exp(-1/4*t)cos(sqrt(15)/4*t) + c2*exp(-1/4*t)sin(sqrt(15)/4*t)-2sin(t) + 1

Put x= 0 into that and its derivative.
y(0)= c1+ 1= 0 so c1= -1.

You do the derivative.
 

What is the Method of Undetermined Coefficients?

The Method of Undetermined Coefficients is a technique used in mathematics and physics to find the particular solution of a non-homogeneous linear differential equation. It is used when the non-homogeneous term has a known form, such as a polynomial, exponential, or trigonometric function.

When is the Method of Undetermined Coefficients used?

The Method of Undetermined Coefficients is used when solving a non-homogeneous linear differential equation with a known non-homogeneous term. It is not applicable to non-linear or non-constant coefficient equations.

How does the Method of Undetermined Coefficients work?

The method involves guessing a particular solution based on the form of the non-homogeneous term. This guess is then substituted into the original differential equation, and the coefficients are solved for using algebraic techniques.

What are the limitations of the Method of Undetermined Coefficients?

The method only works for certain forms of non-homogeneous terms, such as polynomials, exponentials, and trigonometric functions. It also does not work for non-linear or non-constant coefficient equations.

Are there any tips for making a good guess in the Method of Undetermined Coefficients?

Yes, there are some tips that can help in making a good guess. For example, if the non-homogeneous term is a polynomial of degree n, the particular solution should be a polynomial of degree n with undetermined coefficients. Also, if the non-homogeneous term involves trigonometric functions, the guess should include all possible trigonometric functions with undetermined coefficients.

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