The Auxiliary Equations
4
The Auxiliary Equations: Distinct Roots Any linear homogeneous diferential equation with constant coecients, a o d n y dx n +a 1 d n−1 y dx n−1 + …+a n−1 dy dx +a n y =0 (1) may be written in the or f ( D ) y =0 (2) where f (D) is a linear diferential operator. As we saw in the preceding chapter, i m is any root o the algebraic equati0n f(m) = 0, then f ( D ) e mx =0, which means simply that y =e mx is a solution o equation (2). he equation f ( m ) =0 (3) is called the auxiliary equation associated with (!) and (2). he au"ilia ry equation or (!) is o degre e n. #e t its ro ots be m ! , m 2 , $, m n . % these roots are all real and distinct, then the n solutions y 1 =exp ( m 1 x ) , y 2 =exp ( m 2 x ) ,…, y n =exp ( m n x ) , are linearly independen t and the general solution o (!) can be written at once. %t is y 1 =c 1 exp ( m 1 x ) +c 2 exp ( m 2 x ) +…+c n exp ( m n x ) , in which c ! , c 2 , $, c n are arbitrary constants.
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Transcript of The Auxiliary Equations
8/19/2019 The Auxiliary Equations
http://slidepdf.com/reader/full/the-auxiliary-equations 1/4
The Auxiliary Equations: Distinct Roots
Any linear homogeneous diferential equation with constant
coecients,
ao
dn y
dxn +a1
dn−1
y
dxn−1
+…+an−1
dy
dx
+an y=0
(1)
may be written in the or
f ( D ) y=0 (2)
where f (D) is a linear diferential operator. As we saw in the
preceding chapter, i m is any root o the algebraic equati0n f(m)
= 0, thenf ( D ) e
mx=0,
which means simply that y=emx
is a solution o equation (2). he
equation
f ( m )=0 (3)
is called the auxiliary equation associated with (!) and (2).
he au"iliary equation or (!) is o degree n. #et its roots be m!,
m2, $, mn. % these roots are all real and distinct, then the n
solutions
y1=exp (m1 x ) , y2=exp ( m2 x ) ,…, yn=exp( mn x ) ,
are linearly independent and the general solution o (!) can be
written at once. %t is
y1=c1 exp (m1 x )+c2 exp (m2 x )+…+cn exp (mn x ) ,
in which c!, c2, $, cn are arbitrary constants.
8/19/2019 The Auxiliary Equations
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EXAMPLE 1
&ol'e the equation
d2
y
dx2−3
dy
dx+2 y=0
irst, write the au"iliary equations
m2−3m+2=0
y actoring
(m−1 ) (m−2 )=0
m1=1,m2=2,
he roots are real and distinct. hus e x∧e2 x are solutions and the
general solution may be written
y=c1 e x+c2 e
2 x
EXAMPLE 2:
ind the general solution to the equation
y'' −2 y
' −8 y=0
&olution*
he associated au"iliary equation is
y2−2 y−8=0
8/19/2019 The Auxiliary Equations
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y actoring
( y−4 ) ( y+2 )=0
y=4, y=−2
he undamental solution set is {e4 x
, e−2 x
} . hus, the general
solution is
y ( x )=c1e4 x+c2 e
−2 x
EXAMPLE 3
&ol'e the equation
d3 y
dx3−4
d2 y
dx2+
dy
dx+6 y=0
irst, write the au"iliary equations
m3−4 m
2+m+6=0
+e obser'e that m -! is a root o this equation. y synthetic
di'ision we obtain the actoriation
(m+1 ) (m2−5m+6 )=0
(m+1 ) (m−2 ) (m−3 )=0
hus the roots are distinct real numbers
m1=−1,m2=2,m3=3,
and the general solution is
y=c1e− x+c2e
2 x+c3 e
3 x
Exercise:
8/19/2019 The Auxiliary Equations
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ind the general solution. +hen the operator / is used, it is
implied that the independence
!) ( D2+2 D−3 ) y=0
2) ( D2+2 D ) y=0
) ( D3
+3 D2
−4 D ) y=0
1) ( D3−2 D
2−5 D+6 ) y=0
repared by*Ms MARR! "#R$%ELLE # RA&'LA
3A45d
