10250 2 பயம

൧൧31 -8

Answer six quéstions = one full question from each Module. ~~

Module - 1

13. दे Reduce to first order and hence solve the ODE

14. a) Show that the functions x and x In (x) are linearly independent
(use Wronskian). Hence form an ODE for the given basis x, x In (x).

i) ೫! + 0/)3 cos y = 0 and
॥) 2xy" = 30.
‏ل‎ Solve the IVP #॥ - 2y! + 5y = 0, y (0) = - 3, y! (0) = 1.
OR

0) Solve the IV Py'' + 0.2 y'+ 4.01 y =0, y (0) = 0, y! (0) =2.

Module - 2

15. ഒ). Solve the differential equation (D + 1(2 = ೫20".

16.

0) Solve the differential equation (1303 + 3x2D? + xD + 1)y=x + logx.
OR

a) Solve the differential equation (೧2 + 1)y = x2e* + sinx.

© Solve the differential equation (x + 1)ಸ/ + (x + 1)y!—y = 2 sin log (x + 1).

Module - 3

1 , 0> <> 1

: 8 + ர்‌ f x)=
a) Find the Fourier Series of f (x) ட, മ

' ‏زم‎ Find the Fourier cosine series of f (x) =x (7 —x) 11044 ௩.

OR

8--a)_Expand f (x) = e™ in (- 1, ൧) as a Fourier Series.

छो Find the half range sine series of f (x) = x sinx ‏أ‎ 0 > > > അ.

11

11

11

11

11

11