1000 Solved Problems in Modern Physics

(Romina) #1

62 1 Mathematical Physics


dy
dx

+Py=Q (2)

dy
dx

+

y(x+1)
x

=9(3)

Lety=Uz (4)
dy
dx

=

Udz
dx

+

zdU
dx

(5)

Substituting (4) and (5) in (3)
Udz
dx

+

(

dU
dx

+

U(x+1)
x

)

z=9(6)

Now to determineU, we place the coefficients ofzequal to zero. This gives
dU
dx

+

U(x+1)
x

= 0

dU
U

=−

(

1 +

1

x

)

dx
Integrating, lnU=−x−lnxor
U=e−x/x (7)

As the term inzdrops off, Eq. (6) becomes

U

dz
dx

=9(8)

EliminatingUbetween (7) and (8)

dz= 9 xexdx

Integratingz= 9


xexdx= 9 ex(x−1) (9)

SubstitutingUandziny=Uz,

y=

9(x−1)
x

1.65 d


(^2) y
dx^2


+

dy
dx

− 2 y=2 cosh 2x (1)

The complimentary solution is found from

d^2 y
dx^2

+

dy
dx

− 2 y= 0

D^2 +D− 2 = 0
(D−1)(D+2)= 0
D= 1 ,− 2
Y=U=C 1 ex+C 2 e−^2 x (2)
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