|TIME: 3 HOURS|
|(8)||1.||Solve the following initial value problems:|
|(a)||4x dy - y dx = x2dy, y(5) = 51/4|
|(b)||(x3 + y3)dx - 3xy2dy = 0, y(2) = 21/3|
|(15)||2.||Find the general solution for the following equations:|
|(a)||y'' - 3y' + 2y = ex|
|(b)||y'' + 9y = 32x cosx|
|(c)||y'' + y' - 2y = 2(1 + x - x2) + 4e2x|
Find the general solution of (use the method of
variation of parameters):
y'' - 2y' = ex sinx.
Find the general solution of (use the
method of power series):
y'' + xy' + y = 0.
|(12)||5.||Test the following series for convergence or divergence:|
||Determine whether the series
absolutely convergent, conditionally convergent or divergent. Give
||Find the interval of convergence of the series
x + x
4 + x9 + x16 + x25 + ... .
||Give the first four non-zero terms in the expansion of
x = e-t, y =
||Find the total length of the astroid
where C consists of
the arc C1 of the parabola y = x2 from (0,0) to (1,1)
followed by the line segment C2 from (1,1) to (0,3).
is a conservative field.
Show that div
where C is the closed curve of the region bounded by y =
x2 and y = x. Use Green's Theorem to set this as a double
integral and then evaluate this double integral.
and S is the surface
of the cube bounded by
x = 0, x = 1, y = 0, y = 1, z = 0, z = 1.
Use divergence theorem to set this as a triple integral and then
evaluate this triple integral.