a) Solve $(1+y^2)dx = (\tan^{-1}y - x)dy$ using Leibnitz linear method.

b) Solve $(e^y+1)\cos x dx + e^y \sin x dy = 0$.

a) Solve $(D^2 - 4D + 3)y = \cos 2x$.

b) Show that $J_{\frac{1}{2}}(x) = \sqrt{\frac{2}{\pi x}}\sin x$.

Solve $(D^2+9)y = \tan 3x$ by using method of variation of parameters.

a) Solve the partial differential equation $(x-y)p + (x+y)q = 2xz$.

b) Solve $(p^2+q^2)y = qz$ by using Charpit's method.

a) Solve $(D^2+4DD'-5D'^2)Z = \sin(2x+3y)$.

b) Determine p so that the function $f(z) = \frac{1}{2}\log(x^2+y^2) + i \tan^{-1}(\frac{px}{y})$ is analytic function.

a) Show that the function $u(x,y) = e^x \cos y$ is Harmonic. Determine it's Harmonic conjugate.

b) Find the residue of $\frac{Ze^z}{(Z-1)^3}$ at it's pole.

Verify Gauss divergence theorem for $\vec{F} = x^3\hat{i} + y^3\hat{j} + z^3\hat{k}$ taken over the cube bounded by $x=0, x=a, y=0, y=a, z=0, z=a$.

a) Prove that $curl(r^n \vec{r}) = \vec{0}$.

b) Write short note on:

i) Cauchy Riemann equations

ii) Stokes theorem