# Method of undetermined coefficients

• cloud18
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
cloud18
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?

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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