Solving y + y = 2/sin(x) using Undetermined Coefficients

In summary, the conversation is about solving the differential equation y" + y = 2/sin(x) and the different methods that can be used to solve it, such as variation of parameters and undetermined coefficients. There is also a discussion about finding a particular solution for sin(x) and the use of exponential functions in solving differential equations. The final solution provided by one of the speakers is confirmed to be correct by differentiating and substituting back into the equation.
  • #1
cimmerian
15
0

Homework Statement



y" + y = 2/sin(x)

solve for y

Homework Equations



I tried to use variation of parameters to solve this but I don't know how to check it.

The Attempt at a Solution



y = -2xcosx + (constant)cosx + 2ln(sin(x))sinx + (constant)sinx

How do I do this using Undetermined coefficients? I can't find a basis for the null space of 2/sinx
 
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  • #2
I don't think there exists a PI for sin(x), you may need to use variation of parameters to solve the problem since you get two solutions for the homogene\eous equation.
 
  • #3
Of course there is.

Anytime you see a sin or cos in differential equations theory you can rewrite it as an exponential. But for an actual particular solution, you can guess a linear combination of sin and cos.
 
  • #4
cimmerian said:

The Attempt at a Solution



y = -2xcosx + (constant)cosx + 2ln(sin(x))sinx + (constant)sinx

This solution is correct,you can check it by differentiating it and subbing it back into the DE.

snipez90 said:
Of course there is.

Anytime you see a sin or cos in differential equations theory you can rewrite it as an exponential. But for an actual particular solution, you can guess a linear combination of sin and cos.

If the sin(x) is in the denominator and you write that in terms of eix and e-ix, you'd have those two on the denominator as well?
 
  • #5
You said a particular integral for sin, and the use of / threw me off. But I agree for csc you would use variation of parameters or Green's functions.
 
  • #6
thanks
 

Related to Solving y + y = 2/sin(x) using Undetermined Coefficients

1. What are undetermined coefficients?

Undetermined coefficients are a method used in solving systems of linear equations with multiple unknown variables. It involves assigning variables, known as coefficients, to each unknown variable and using algebraic manipulation to solve for their values.

2. When is the method of undetermined coefficients used?

The method of undetermined coefficients is commonly used when faced with a system of linear equations with multiple unknown variables, where the equations are not linearly independent. It is also used in solving differential equations and in finding particular solutions of non-homogeneous linear equations.

3. How does the method of undetermined coefficients work?

The method of undetermined coefficients works by assigning coefficients to each unknown variable in the system of equations. These coefficients are then substituted into the equations and solved through algebraic manipulation. The resulting values of the coefficients can then be used to solve the system of equations.

4. What are the advantages of using undetermined coefficients?

The method of undetermined coefficients is advantageous because it can be used to solve systems of equations with multiple unknown variables, even if the equations are not linearly independent. It is also a relatively simple and efficient method, compared to other methods such as Gaussian elimination or Cramer's rule.

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

The method of undetermined coefficients is limited in its applicability, as it can only be used for certain types of systems of equations. It also relies on the assumption that the equations are not linearly independent, which may not always be the case. In addition, it may not always yield unique solutions for the unknown variables.

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