Undetermined coefficients

  • Thread starter cimmerian
  • Start date
  • #1
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

Homework Statement





Homework Equations





The Attempt at a Solution

 

Answers and Replies

  • #2
rock.freak667
Homework Helper
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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
1,101
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
rock.freak667
Homework Helper
6,223
31

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.

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
1,101
3
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
15
0
thanks
 

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