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

[cimmerian]

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

 

[rock.freak667]

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.

 

[snipez90]

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.

 

[rock.freak667]

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 e^ix and e^-ix, you'd have those two on the denominator as well?

 

[snipez90]

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.

 

[cimmerian]

thanks