r
CONSTANT COEFFICIENTS :
① Find
Auxiliary Polynomial
② Find roots of the
polynomial :
↳ If there two distinct roots : hi .dz
y= tie
" 't
1- ✗ zetzt
↳ If :X
there is one
repeated root ,
y=
diet 't taztet 't
↳ If the roots :a±ib
conjugates
are
complex
"'
the coslbx) fazed" sinlbx )
y=
NONHOMOGENEOUS DES
Annihilated : DE =
fcx )
↳ Choose I CDTI) such that f. ( x) C- Her CI )
an
operator e.
g
↳ This makes the DE
homogeneous
↳ Solve PE
the
homogeneous
↳ Substitute back into the DE and solve for the unknown constant .
, VARIATION OF PARAMETERS
↳ Find the form
homogeneous
↳ Find solutions yiiyz Ot the homogeneous DE
i.
Yc =
Nyi tdzyz
↳ Use the
following to find Ui and 620T :
Yp =
Uiy , t
6292
(
Yi
yi
'
Yi
ya
'
/ 1%1=1%1%1
↳ Combine both :
y= Yc tyr
+ ✗
=
ay ,
292 1-
Uiyituzyz
DERIVATION :
92
Y
"
t
Aly / taoy =fCx)
has solution 1- ✗
y=gP iyittzyz
Assume Ui
Yi
yp tuzyz
=
' '
Uiyi
/
i.
Yp
=
1- Uiyi 1-
Uiy , tltryz
Impose Uilyl tUz1Yz=0
Uiyi 4292
/ '
t
yp
'
i. =
CONSTANT COEFFICIENTS :
① Find
Auxiliary Polynomial
② Find roots of the
polynomial :
↳ If there two distinct roots : hi .dz
y= tie
" 't
1- ✗ zetzt
↳ If :X
there is one
repeated root ,
y=
diet 't taztet 't
↳ If the roots :a±ib
conjugates
are
complex
"'
the coslbx) fazed" sinlbx )
y=
NONHOMOGENEOUS DES
Annihilated : DE =
fcx )
↳ Choose I CDTI) such that f. ( x) C- Her CI )
an
operator e.
g
↳ This makes the DE
homogeneous
↳ Solve PE
the
homogeneous
↳ Substitute back into the DE and solve for the unknown constant .
, VARIATION OF PARAMETERS
↳ Find the form
homogeneous
↳ Find solutions yiiyz Ot the homogeneous DE
i.
Yc =
Nyi tdzyz
↳ Use the
following to find Ui and 620T :
Yp =
Uiy , t
6292
(
Yi
yi
'
Yi
ya
'
/ 1%1=1%1%1
↳ Combine both :
y= Yc tyr
+ ✗
=
ay ,
292 1-
Uiyituzyz
DERIVATION :
92
Y
"
t
Aly / taoy =fCx)
has solution 1- ✗
y=gP iyittzyz
Assume Ui
Yi
yp tuzyz
=
' '
Uiyi
/
i.
Yp
=
1- Uiyi 1-
Uiy , tltryz
Impose Uilyl tUz1Yz=0
Uiyi 4292
/ '
t
yp
'
i. =
