## anonymous 4 years ago For $y''+y'-2y=sinx$ $y_p=Acosx+Bsinx$ why is it incorrect to guess: $y_p=Asinx$

1. experimentX

y_p doesn't contain constants of integration.

2. anonymous

@experimentX , what do you mean by that?

3. experimentX

what's the difference between solution of $$y''+y'-2y=0$$ and $$y''+y'-2y=\sin x$$

4. anonymous

one has a particular solution and the other doesn't

5. experimentX

for second order equation, you should have 2 constants of integration ... first one will have two constants .. now if you put another constant to particular solution ... and you will have 3 constants

6. anonymous

could you illustrate that? sorry :S

7. experimentX

|dw:1352301416695:dw|

8. experimentX

guessing particular solution is quite difficult job ... easy when you have polynomials one the RHS. probably you are seeking this method http://en.wikipedia.org/wiki/Method_of_undetermined_coefficients

9. anonymous

makes sense. Thanks!

10. experimentX

the particular solution will be of the form $y_p=A\cos x+B\sin x$ you have to find the values of A and B by plugging into equation. I probably understand your Q quite clearly now. ----------------------- plug $$y_p = A \sin (x)$$ into DE, you have $y'' = \sin(x) \text{ and } y'(x) = \cos(x)$ no matter what the value of constant's ... there is no Cos(x) on RHS ... you cannot simply have Yp = A sin(x)

11. anonymous

12. experimentX

honestly i would prefer "Reduction of Order" or "Annihilation Operator" method .. they are quite faster than "Undetermined coefficients"

13. anonymous

Example 1's y_c is that in the above attachment, but how is that relevant when example 1 is y''+y'-2y=x^2 ??? I guess I'll just stick with "seeing patterns" and memorization :P

14. experimentX
15. anonymous

Oh where would I be without Paul's online notes? http://tutorial.math.lamar.edu/Classes/DE/UndeterminedCoefficients.aspx In example 3 he does what I would have guessed y_p=Asin(2t) but then he goes on to explain why that guess is flawed. something that @experimentX and @TuringTest have been trying to explain to me all along :S Thank's guys :)

16. experimentX

sorry .. went offline :(