490 Higher Engineering Mathematics
Now try the following exercise
Exercise 191 Further problems on
differential equations of the form
a
d^2 y
dx^2
+b
dy
dx
+cy=f(x)wheref(x)is a sine or
cosine function
In Problems 1 to 3, find the general solutionsof the
given differential equations.
- 2
d^2 y
dx^2
−
dy
dx
− 3 y=25sin2x
[
y=Ae
3
2 x+Be−x
−^15 (11sin2x−2cos2x)
]
2.
d^2 y
dx^2
− 4
dy
dx
+ 4 y=5cosx
[
y=(Ax+B)e^2 x−^45 sinx+^35 cosx
]
3.
d^2 y
dx^2
+y=4cosx
[y=Acosx+Bsinx+ 2 xsinx]
- Find the particular solution of the differen-
tialequation
d^2 y
dx^2
− 3
dy
dx
− 4 y=3sinx;when
x=0,y=0and
dy
dx
=0.
⎡
⎢
⎢
⎣
y=
1
170
(6e^4 x−51e−x)
−
1
34
(15sinx−9cosx)
⎤
⎥
⎥
⎦
- A differential equation representing the
motion of a body is
d^2 y
dt^2
+n^2 y=ksinpt,
wherek,nandpare constants. Solve the equa-
tion (givenn=0andp^2 =n^2 ) given that when
t=0,y=
dy
dt
=0.
[
y=
k
n^2 −p^2
(
sinpt−
p
n
sinnt
)]
- The motion of a vibrating mass is given by
d^2 y
dt^2
+ 8
dy
dt
+ 20 y=300sin4t. Show that the
general solution of the differential equation is
given by:
y=e−^4 t(Acos 2t+Bsin2t)
+
15
13
(sin 4t−8cos4t)
- L
d^2 q
dt^2
+R
dq
dt
+
1
C
q=V 0 sinωtrepresents the
variation of capacitor charge in an elec-
tric circuit. Determine an expression for
q at time tseconds given that R= 40 ,
L= 0 .02H, C= 50 × 10 −^6 F, V 0 = 540 .8V
and ω=200rad/s and given the boundary
conditionsthat whent=0,q=0and
dq
dt
= 4. 8
[
q=( 10 t+ 0. 01 )e−^1000 t
+ 0 .024sin200t− 0 .010cos200t
]
51.6 Worked problemson
differential equations of the
forma
d^2 y
dx^2
+b
dy
dx
+cy=f(x)
wheref(x)is a sum or a product
Problem 9. Solve
d^2 y
dx^2
+
dy
dx
− 6 y= 12 x−50sinx.
Using the procedure of Section 51.2:
(i)
d^2 y
dx^2
+
dy
dx
− 6 y= 12 x−50sinx in D-operator
form is
(D^2 +D− 6 )y= 12 x−50sinx
(ii) The auxiliary equation is(m^2 +m− 6 )=0, from
which,
(m− 2 )(m+ 3 )= 0 ,
i.e. m=2orm=− 3
(iii) Since the roots are real and different, the C.F.,
u=Ae^2 x+Be−^3 x.
(iv) Since the right hand side of the given differential
equation is the sum of a polynomial and a sine
function let the P.I.v=ax+b+csinx+dcosx
(see Table 51.1(e)).