082-Engineering-Mathematics-Anthony-Croft-Robert-Davison-Martin-Hargreaves-James-Flint-Edisi-5-2017

568 Chapter 19 Ordinary differential equations I
If the auxiliary equation has two equal roots,k 1
, the complementary function is
y =Ae k 1 x +Bxe k 1 x
Example19.25 Obtain the general solution of the equation
d 2 y
dx 2 +8dy dx +16y=0
Solution As before, a trialsolution of the formy = e kx yields an auxiliary equation:
k 2 +8k+16=0
This equation factorizes so that
(k+4)(k+4)=0
and we obtain equal roots, that is k = −4 (twice). If we proceed as before, writing
y 1
(x) = e −4x , y 2
(x) = e −4x , it is clear that the two solutions are not independent. To
apply Property 1 we need to find a second independent solution. Using the result of
Example 19.24 we conclude that, because the roots of the auxiliary equation are equal,
the second independent solution isy 2
=xe −4x . The general solution isthen
y(x) =Ae −4x +Bxe −4x
EXERCISES19.6.1
1 Obtainthe general solutions,that isthe
complementary functions,ofthe following
homogeneous equations:
(a)
(b)
(c)
(d)
(e)
(f)
(g)
d 2 y
dx 2 −3dy dx +2y=0
d 2 y
dx 2 +7dy dx +6y=0
d 2 x
dt 2 +5dx dt +6x=0
d 2 y
dt 2 +2dy dt +y=0
d 2 y
dx 2 −4dy dx +4y=0
d 2 y
dt 2 + dy
dt +8y=0
d 2 y
dx 2 −2dy dx +y=0
(h)
(i)
(j)
(k)
(l)
d 2 y
dt 2 + dy
dt +5y=0
d 2 y
dx 2 + dy
dx −2y=0
d 2 y
dx 2 +9y=0
d 2 y
dx 2 −2dy dx = 0
d 2 x
dt 2 −16x=0
2 Findthe auxiliaryequation forthe differential
equation
L d2 i
dt 2 +Rdi dt + 1 C i = 0
Hence write down the complementary function.
3 Findthe complementaryfunctionofthe equation
d 2 y
dx 2 + dy
dx +y=0