Diff. Eq. : Undetermined Coefficents

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In summary, There were two problems discussed in this conversation. The first involved an ODE with the guess of y=A[tsin(2t)]+B[tcos2t]. The person only got part of the answer correct and then tried another guess, y= A[t^2(sin(2t)]+ B[t^2(cos(2t)] + C[tsin(2t)]+D[tcos2t], which gave the same answer as before plus some incorrect answers. The second problem involved an ODE with the guess of Ae^t+Bte^(-t). The person got an answer of -2A=1/2 and -3B=1/2, but these were also incorrect.
  • #1
NINHARDCOREFAN
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I have 2 problems that I'm not getting the right answer to:

y"+4y=3sin(2t)

I chose my guess to be: y=A[tsin(2t)]+B[tcos2t]
with this guess I'm getting only part of the answer right

I also tred this guess: y= A[t^2(sin(2t)]+ B[t^2(cos(2t)] + C[tsin(2t)]+D[tcos2t]

I got the same answer as above + some other weird answers


y"-y'-2y=(e^t+e^(-t))/2

I made this guess: Ae^t+Bte^(-t)

I got my answer as -2A=1/2 and -3B=1/2

However these were also wrong. Anything wrong with the guesses?
 
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  • #2
What were the fundamental (complimentary) set of solutions for each ODE?
 
  • #3
Answers...

The first one: Asin(2t)+Bcos(2t)-(1/8)sin(2t)-(3/4)tcost(2t)
The second one : Ae^-t+Be^-2t+(1/6)te^(2t)+(1/8)e^(-2t)
 

FAQ: Diff. Eq. : Undetermined Coefficents

What is the purpose of using undetermined coefficients in differential equations?

The purpose of using undetermined coefficients in differential equations is to find a particular solution that satisfies the given equation by assuming a form for the solution and solving for the unknown coefficients. This method is often used for non-homogeneous linear equations with constant coefficients.

How do you determine the form of the particular solution when using undetermined coefficients?

The form of the particular solution is determined by the non-homogeneous term of the differential equation. If the non-homogeneous term is a polynomial of degree n, then the particular solution will be a polynomial of degree n with undetermined coefficients.

Can undetermined coefficients be used for all types of differential equations?

No, undetermined coefficients can only be used for non-homogeneous linear equations with constant coefficients. For other types of differential equations, other methods such as variation of parameters or Laplace transforms may be used.

How do you solve for the undetermined coefficients in a differential equation?

To solve for the undetermined coefficients, substitute the assumed form of the particular solution into the original differential equation and solve for the coefficients by equating coefficients of like terms. This will result in a system of equations that can be solved to find the values of the coefficients.

What is the difference between undetermined coefficients and variation of parameters?

Both methods are used to find particular solutions for non-homogeneous linear equations. The main difference is that undetermined coefficients assumes a form for the particular solution, while variation of parameters uses an integral to find a general solution and then applies boundary conditions to determine the particular solution. Undetermined coefficients is typically easier to use but may not always work for more complex equations, while variation of parameters is more general but can be more time-consuming.

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