Method of Undetermined Coefficients

In summary, the form for the solution of a second order ODE with a right hand side of x (sin x + 2) is (Ax + B) sin(x) + (Cx + D) cos(x) + Ex + F. However, it is impossible to determine if this form is correct without knowing the homogeneous equation and its solutions. If any of the given solutions are already present in the homogeneous equation, the form of the solution will need to be adjusted accordingly.
  • #1
jmg498
8
0
I am looking for the form of a solution for a second order ODE with right hand side: x (sin x + 2)

I'm thinking the form would be (Ax + B) sin(x) + (Cx + D) cos(x) + Ex + F. Does this seem correct?

Thanks for any help or suggestions!
 
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  • #2
Have you tried it? That would tell you whether it is correct or not! If it is correct, you will get an answer. If it is not correct, you won't.

Actually, it is impossible for us to tell because you haven't given us the homogeneous equation or its solutions. If sin(x) or cos(x) or x sin(x) or x cos(x) are already solutions to the homogenous equation, you will need to multiply (Ax+B)sin(x) and (Cx+ D)cos(x) by x again. If x or a constant is already a solution to the homogeneous equation, you will need to Ex2+ Fx.
 

1. What is the method of undetermined coefficients?

The method of undetermined coefficients is a mathematical technique used to solve nonhomogeneous linear differential equations. It involves finding a particular solution to the equation by guessing a form for the solution and determining the unknown coefficients through substitution.

2. When is the method of undetermined coefficients used?

This method is used when the nonhomogeneous linear differential equation has constant coefficients and the nonhomogeneous term is a sum of exponential, sine, cosine, or polynomial functions. It is not applicable to equations with variable coefficients or non-polynomial nonhomogeneous terms.

3. What is the process for using the method of undetermined coefficients?

The process involves the following steps: 1) Identify the form of the particular solution based on the nonhomogeneous term. 2) Substitute the form into the original equation and solve for the unknown coefficients. 3) Combine the particular solution with the complementary solution (solution to the corresponding homogeneous equation) to find the general solution.

4. How is the form of the particular solution determined?

The form of the particular solution is determined by the type of nonhomogeneous term in the equation. For example, if the nonhomogeneous term is a constant, the particular solution will be a constant. If it is a polynomial, the particular solution will be a polynomial of the same degree.

5. What are the limitations of the method of undetermined coefficients?

The method of undetermined coefficients is limited to solving certain types of nonhomogeneous linear differential equations with constant coefficients. It cannot be used for equations with variable coefficients or non-polynomial nonhomogeneous terms. Additionally, the method may fail if the guessed form of the particular solution is a solution to the corresponding homogeneous equation.

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