10.

11.

12.

13.

14.

15.

16.

17.

18.

108957

2 C 20648

Find the Laplace transform of' the function sin (at).

n!

Find the inverse Laplace transform of (नोन्न where s > a.

Let w,(¢) be unit step function and L(f (t))=F(s). Show that :

L(u, (t)f(t-c)) =e*F(s).

Find the inverse Laplace transform of the following function by using the convolution theorem
--‡-

st (83 + 1)

Solve the boundary value problem :

y"+y =0, (0) =0, y(z)=0.

Define an even function and show that if f (x) is an even function then :

L L
[ f (x)dx = 2] f(x) de
-L 0

Define the following partial differential equations :
(a) heat conduction equation.
(b) one-dimensional wave equation.

(10 x 3 = 30 marks)
Section B

Answer at least five questions.

Each question carries 6 marks.

All questions can be attended.
Overall Ceiling 30.

Let +] (४) € 8 solution of y+ p(t)y=0 andlet y(t) be a solution of y' + p(t) y= g(t).

Show that ೨'(1) 7೫] (1) * ೫ (1) 18 also a solution of equation y+ p(t)y=s8(t).

Find the value of 6 for which the following equation is exact, and then solve it using that value
of b.

(xy? + bxy)+ (x+ y)x?y =0.
Solve the initial value problem

4" + 4+ = +3९४,)(0) = 0, 40) = 2.

108957