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01000MA102032103

PART 0
Answer six questions, one full question from each module

Module I
Find a basis of solutions of the ODE, xy" + xy’ — 4) = 0. Given y, = x? is
one solution.
Determine all possible solutions to the initial value problem, ‏کر‎ = 1 + y?,
y(0) = 0 in the interval |x| < 3,and |y| < 2
OR
Verify by substitution that y, = e~*cosx and yz = ടാ are the solutions
of the given ODE and then solve the initial value problem, y" + 2y’ + 29 = 0,
y(0) = 0,y'(0) = 15
Find the general solution 08) ೫ — 2y'” + 2y” —2y’+y=0
Module 11
Solve, by the method of variation of parameters , (D2 + 1)y = cosecx
Solve,(D? — D* - 60)» =x? + 1
OR
Solve,(D? - 2D + 5)) ടാ
dy

24 பய 6० _
Solve, x om 4x73 + 6 ‏حدس‎ 4

Module III

0, Spee ‏پر‎
‎sinx, O
OR

7%, நற்கு
m(2—x), 1
Find the Fourier series of f (x) = {

Find the Fourier series of f (x) = {

Obtain the half range Fourier cosine series of f(x) = (x — 1)?,0
1 1 1 72
Show that 12 + 22 + 32 −⊦⋅⋅− =

Module IV
Solve, 7 — 45 + 46 = ഓനാ
Solve, (22 - ‏م(مز‎ + (ॐ + ‏ع2- = ورج‎ - 9
OR
Solve, (02 — 2DD' - 15D’*)z = 12xy

Find the PDE of all planes cutting equal intercepts from the X and Y axes.

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