a) Solve \( (1+y^2)dx = (\tan^{-1}y-x)dy \).

b) Solve \( (D^2+3D+2)y = \sin 3x \).

a) Solve the simultaneous equations \( \frac{dx}{dt} - 7x + y = 0 \) and \( \frac{dy}{dt} - 2x - 5y = 0 \).

b) Solve by the method of variation of parameter \( (D^2+1)y=x \).

a) Solve \( (1+x)^2 \frac{d^2y}{dx^2} + (1+x)\frac{dy}{dx} + y = \cos \log(1+x) \).

b) Show that \( J_n(-x) = (-1)^n J_n(x) \) when n is positive or negative integer.

a) Solve by Charpit's method, \( px+qy=pq \).

b) Solve the Partial differential equation \( (D^3 - 4D^2D' + 4DD'^2)Z = \cos(2x+y) \).

a) Construct a partial differential equation from the relation \( f(x^2+y^2+z^2, z^2-2xy) = 0 \).

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

a) Determine P such that the function \( f(z) = \frac{1}{2}\log(x^2+y^2) + i \tan^{-1}(\frac{px}{y}) \) be an analytic function.

b) Evaluate using Cauchy's theorem \( \oint_C \frac{z^2e^{-z}}{(z-1)^2}dz \) where c is \( |z-1|=\frac{1}{2} \).

a) Find the poles and residues at each pole of \( \frac{e^z}{z^2+1} \).

b) Find the directional derivative of \( \emptyset = x^2yz + 4xz^2 \) at \( (1, -2, -1) \) in the direction of \( 2\hat{i} - \hat{j} - 2\hat{k} \).

Verify Green's theorem for \( \oint_C [(xy+y^2)dx + x^2dy] \) where C is the boundary by \( y=x \) and \( y=x^2 \).