Advanced Engineering Mathematics — Winter 2019

(i) Solve the I. V. P : y '' 4 y ' 4 y  0 , y ( 0 )  3 & y '( 0 )  (ii) Find (a) L  e t ( s i n 3 t  t 3 )  (b) L  1  s s2  7 03

Determine the singular points of the differential equation x ( x  1 ) 2 y '' ( 2 x  1 ) y ' x 2 y  and classify them as regular or irregular.

Compute

5 ,3   02 (ii) Define (1) Error Function (2) Beta Function

   s(sa)3 

Solve (1  x ) y d x  (1  y ) x d y  0

Solve ( D 2  5 D  6 ) y  s i n 3 x  1  State the convolution theorem and apply it to evaluate L1

Using Laplace Transformation, Solve y ''  6 y  1 , y ( 0 )  2 , y '( 0 )  0

Find the series solution of y '' x 2 y  0 in power of x .

Find the Laplace transform of f ( t )   , , 0  t t  2

Find the power series solution of y'  2xy.

0, 2 x 0 Obtain Fourier series of the Function f(x) 1, 0 x 

Q.3 Find the Inverse Laplace Transform of

(6 s e2   2 s 4 )03

Obtain Fourier series of the Function f(x)  x |x| ,  x 

dy Solve  y tanx sin2x dx

Find a sine series for f ( x )  e x in 0 x . u u By the Method of Separation of variables , solve 2   u where

x t u(x,0)4e3x

Solve yexdx(2yex)dy 0, y(0)  1

Find a cosine series for2 f ( x )  x 2 in 0 x    .

Using Undetermined co-efficient method , solve the differential equation y '' y ' 6 y  6 x  3 x 2  6 x 307

Solve z  pxqyp2q2 Find (1) (b) L  t0 e  t c o s u d u  (2) L  1  s 22  s2  s2  1 0 

Find the general solution of P. D. E : (x2  yz) p(y2  zx)q z2  xy

Form a Partial differential equation from f ( x y  z 2 , x  y  z )  0 Find (1) (b) L  t0 t0 s i n a u d u d u  (2) L  1  s s  4 s  

Using Method of Variation of parameters, Solve ( D 2  2 D  1 ) y  3 x32 e x07