1000 Solved Problems in Modern Physics

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)