a) Solve \( x\frac{dy}{dx} + y = x^3y^6 \) using Bernoulli's.

b) Solve the differential equation \( (xe^y+2y)\frac{dy}{dx} + y e^y = 0 \) using Exact method.

a) Solve \( (D^2-6D+13)y = 8e^{3x}\sin 2x \).

b) Show that \( \frac{d}{dx}[x^n J_n(x)] = x^n J_{n-1}(x) \).

Solve \( (D^2+1)y = x \sin x \) using variation of parameters.

a) Form the partial differential equation (By eliminating the arbitrary functions) from \( Z=(x+y)\phi(x^2-y^2) \).

b) Solve \( (D^2-DD'-6D'^2)Z = xy \).

a) Solve the partial differential equation \( yp-xp=z \).

b) Show that \( u = e^{-x}(x \sin y - y \cos y) \) is Harmonic.

a) Evaluate \( \int_{(0,0)}^{(1,1)} (3x^2+4xy+ix^2)dz \) along \( y=x^2 \).

b) Find the Poles and Residues at each pole of \( f(z) = \frac{\sin^2 z}{(z-\frac{\pi}{6})^2} \).

Verify Green's theorem in the plane for \( \oint_C [(x^2-xy^3)dx+(y^2-2xy)dy] \) where C is a square with vertices (0,0), (2,0), (2,2), (0,2).

a) Find the directional derivative of \( f(x,y,z) = xy^2+yz^3 \) at point (2, -1, 1) in the direction of the vector \( \hat{i}+2\hat{j}+2\hat{k} \).

b) Write short note on:

i) Cauchy's integral formula

ii) Solenoidal and Irrotational