a) Solve \( \cos x \frac{dy}{dx} = y(\sin x - y) \) using Bernoulli's.

b) Solve the Linear differential equation \( \sin 2x \frac{dy}{dx} - y = \tan x \).

a) Solve \( (r + \sin\theta - \cos\theta)dr + r(\sin\theta + \cos\theta)d\theta = 0 \).

b) Solve the differential equation. \( (D^3 - 7D^2 + 14D - 8)y = e^x \cos 2x \).

Solve \( (D^2 + 4)y = \tan 2x \) by using method of variation of parameters.

a) Show that \( \frac{\vec{r}}{r^3} \) is solenoidal.

b) Show that the vector \( (x^2 - yz)\hat{i} + (y^2 - zx)\hat{j} + (z^2 - xy)\hat{k} \) is irrotational. Find it's scalar potential.

Verify Green's theorem for \( \oint_C [(3x^2 - 8y^2) dx + (4y - 6xy) dy] \). Where C is the region bounded by $x=0, y=0$ and $x+y=1$.

a) Show that $f(Z) = Z\bar{Z}$ is differentiable but not analytic at origin.

b) Show that $u(x,y) = e^{-2x} \sin 2y$ is harmonic and determine it's Harmonic conjugate.

a) By Residue theorem, Evaluate \( \oint_C \frac{\tan z}{z^2-1} dz \), where C:|Z|=2.

b) Using Cauchy integral theorem, to evaluate the integral \( \oint_C \frac{e^{2z}}{(z-1)^2(z-3)} dz \), where C is the circle |Z|=2.

a) Solve $x^2p^2 + y^2q^2 = z^2$.

b) Solve $(D^2 - 4DD' + 4D'^2)Z = \cos(x-2y)$.