1000 Solved Problems in Modern Physics

(Romina) #1

1.3 Solutions 65


d^4 y
dx^4

+

4d^2 y
dx^2

=−8 cos(2x)(2)

Multiply (1) by 4 and add to (2),
d^4 y
dx^4

+

8d^2 y
dx^2

+ 16 y= 0

D^4 + 8 D^2 + 16 = 0
(D^2 +4)^2 = 0
D=± 2 i
y=C 1 sin 2x+C 3 xsin 2x+C 2 cos 2x+C 4 xcos 2x
=U+C 3 xsin 2x+C 4 xcos 2x
=U+V
Y=V=C 3 xsin 2x+C 4 xcos 2x (3)
Use (3) in (1) and compare the coefficients of sin 2xand cos 2xto find
C 3 = 1 /2 andC 4 =0. Thus the complete solution is

y=C 1 sin 2x+C 2 cos 2x+

1

2

xsin 2x

1.69

dy
dx

+

3 y
x+ 2

=x+ 2

This equation is of the form
dy
dx

+yp(x)=Q(x)

withP=x+^32 andQ=x+ 2
The solution is obtained from
yexp

(∫

p(x)dx=

[∫

Q(x)exp

(∫

p(x)dx

)]

dx+C

Now


p(x)dx= 3


dx
x+ 2

=3ln(x+2)=ln(x+2)^3

∴yexp (ln(x+2)^3 )=

[∫

(x+2) exp (ln (x+2)^3 )

]

dx+C

y(x+2)^3 =


(x+2)^4 dx+C

=

(x+2)^5
5

+C

ory=

(x+2)^2
5

+C

y=2 whenx=− 1
ThereforeC=^95
The complete solution is

y=

(x+2)^2
5

+

9

5

=

x^2 + 4 x+ 13
5
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