19.

20.

21.

22.

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24.

28.

26.

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3 C 20648

Use method of variation of parameters find the general solution of :
y"+4y = 8tant,-1/2
Find the solution of the initial value problem :

20" + 1 + 20 = 6( - 5), (0) = 0,4"(0) = 0.

here 6(t) denote the unit impulse function.

Using Laplace transform solve the initial value problem :

y"+4y =0, (0) =3, 9'(0) =-1.

Find the co-efficients in the Fourier series for /:

0, -3 < > < -1
7 (८) = 41, -1< < 1
0, 1> >8

Also suppose that f (x + 6) = f (x).
Find the solution of the following heat conduction problem :

1001, 5८८८0 < ४ < 1,/ > 0
u(0,t) = 0,u(1,t) =0,t > 0
u(x,0) =sin(2m)—sin(5nx),0< x <1.
(5 x 6 = 30 marks)
Section C

Answer any two questions.
Each question carries 10 marks.

Find the general solution of the following differential equation using the method of integrating
factors:

dy | 1, _1 //3,
dt 2 2

Draw some representative integral curves of the differential equation and also find the particular
solution whose graph contains the point (0,1).

Find a series solution of the differential equation :

y"+y=0, 1
t
Find the Laplace transform of | sin(t—t)cost dt

Find the temperature u (x, t) at any time in a metal rod 50 cm long, insulated on the sides, which
initially has a uniform temperature of 20°C throughout and whose ends are maintained at 0°C for
allt>0.

(2 x 10 = 20 marks)

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