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00000MA102121804

PART 13
Answer six questions, one full question from each module
Module 1

Solve y" - 22 - 39 = 0, x(-D=e¢y'Q = a
Show that the functions x? and 25 are the basis of solutions of ODE
x?y" —7xy' + 15y 0,

OR
Solve ODE +” — 39" ஆது - ^ = 0.

Solve the ODE xy" + 2y'+ xy = 0. Given that y, = न्ने is a solution,

Module 11
By the method of variation of parameters, solve y" + क = 5९८८,

Solve x?y" — 4xy' + 6) = ८५.

OR
Solve (2x + 3)*y" — 2(2x + 3)y' — 129 = 6%.
Solve ‏“بن‎ + 297 - 3y = e*sinx .
Module 111

Find the Fourier series of f defined by f(x) = e* in (—7,7).
OR

Obtain Fourier series for the function f(x) = 2, -ए < ८
Expand f(x) = cosx as a half range sine-series inO
Module 1V

Find the general solution of x*p + y?q = (x + y)z.

OR
Solve 4r + 125 + 9t = e377,

Solve (02 — DD' - 6D")z = xy.
Module V

2

Using method of separation of variables, solve y*u, ‏عد‎ ಜೃ, = 0.

Find the displacement of a finite string of length / that is fixed at both ends and is

released from rest with an initial displacement of 2 sin (=)

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