• Support PF! Buy your school textbooks, materials and every day products Here!

Differential Annihilator

  • Thread starter Gogeta007
  • Start date
  • #1
23
0

Homework Statement


1. y''' + y'' = 8x^2
2. y'' - 2y' - 3y = 4e^x - 9

Homework Equations


Annihilator for any polinomial D^n+1

annihilator for e functions = e^ax = (D-a)^n


The Attempt at a Solution



1. getting the complementary eq.
values of m = 0, 0, -1
yc= c1e^-x +c2 + xc3

then we multiply both sides by the annihilator
D^3<D^2(D+1)> = 8x^2<D^3>

we get y= c1e^-x +c2 + xc3 + x^2c4 + x^3c5

and from the original eq. the x^2 term has a coefficient of 8
therefore c4 = 8
sice there is no x^3 term then c5 should be zero ?
and what about c4?

the answer on the back says the answer is:
c1+c2x+c3e^-x + 2/3 x^4 + 8/3x^3 + 8x^2

===========

2.values of m=3, 2

yc=c1e^3x + c2e^-x
if we multiply by the annihilator alpha = 1 in this case therefore annihilator would be (D-1)
<D-1><D^2-2D-3>=4e^x-9 <D-1>
so we have
yc=c1e^3x + c2e^-x + c3e^x

how do I go from there?

thank you!
 

Answers and Replies

  • #2
699
5
Use the wronskian and cramer's rule.
 
  • #3
vela
Staff Emeritus
Science Advisor
Homework Helper
Education Advisor
14,620
1,254

Homework Statement


1. y''' + y'' = 8x^2
2. y'' - 2y' - 3y = 4e^x - 9

Homework Equations


Annihilator for any polinomial D^n+1

annihilator for e functions = e^ax = (D-a)^n


The Attempt at a Solution



1. getting the complementary eq.
values of m = 0, 0, -1
yc= c1e^-x +c2 + xc3

then we multiply both sides by the annihilator
D^3<D^2(D+1)> = 8x^2<D^3>

we get y= c1e^-x +c2 + xc3 + x^2c4 + x^3c5
You should have one more power of x.
and from the original eq. the x^2 term has a coefficient of 8
therefore c4 = 8
sice there is no x^3 term then c5 should be zero ?
and what about c4?

the answer on the back says the answer is:
c1+c2x+c3e^-x + 2/3 x^4 + 8/3x^3 + 8x^2
You're thinking y=8x^2, but it's actually y'''+y''=8x^2. You need to plug your trial solution back into the differential equation and then solve for the coefficients.
 
  • #4
vela
Staff Emeritus
Science Advisor
Homework Helper
Education Advisor
14,620
1,254

Homework Statement


2. y'' - 2y' - 3y = 4e^x - 9

2. values of m=3, 2
m=2?

yc=c1e^3x + c2e^-x
if we multiply by the annihilator alpha = 1 in this case therefore annihilator would be (D-1)
<D-1><D^2-2D-3>=4e^x-9 <D-1>
so we have
yc=c1e^3x + c2e^-x + c3e^x

how do I go from there?!
You need a slightly different annihilator since

(D-1)(4 ex-9) = 9

Again, once you get your trial solution, plug it back into the differential equation and then solve for the coefficients.
 
  • #5
23
0
Thank you all. I have one more

y'' + y' + y = xsinx

The annihilator for xsinx is (D2+1)2 which would yield 4 answers that are imaginary (+i,-i,+i,-i)
when you do the particular solution, since the answers are repeating do you put an x before the constant?
so far i have that yc=e-x/2(c_1cos<(sqr3)/2> + c_2sin<(sqr3)/2>)
then the annihilator values for D
yp= c_3cosx + c_4sinx + c_5cosx + c_6sinx
or is it
yp= c_3cosx + c_4sinx + xc_5cosx + xc_6sinx

and if its the latter, to get the coefficients you have to take up to the third derivative. . .using product rules? (...im hoping not) or is there another "easier" way?


thank you
 
Last edited:
  • #6
vela
Staff Emeritus
Science Advisor
Homework Helper
Education Advisor
14,620
1,254
Your complementary solution is wrong. You have a third-order differential equation, so you should have three independent solutions. Yours only has two.

You need the second particular solution you wrote down because you need four linearly independent solutions. You multiply by powers of x to make the repeated solutions independent.
 
  • #7
23
0
my bad. . .it is double prime not triple.
*edited*
thanks for pointing that out
 
  • #8
vela
Staff Emeritus
Science Advisor
Homework Helper
Education Advisor
14,620
1,254
my bad. . .it is double prime not triple. *edited*
Well, at least you only have to differentiate yp twice.

The arguments of the cosine and sine in the complementary solution should have an x in it.
 

Related Threads on Differential Annihilator

Replies
10
Views
5K
  • Last Post
Replies
0
Views
1K
  • Last Post
Replies
2
Views
2K
  • Last Post
Replies
2
Views
1K
  • Last Post
Replies
1
Views
3K
  • Last Post
Replies
8
Views
4K
  • Last Post
Replies
2
Views
3K
  • Last Post
Replies
18
Views
4K
  • Last Post
Replies
2
Views
2K
  • Last Post
Replies
18
Views
3K
Top