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

b) Solve $(D^2-5D+6)y = 4e^x+5$.

a) Solve $x^2\frac{d^2y}{dx^2} + 5x\frac{dy}{dx} + 4y = x\log x$.

b) Solve in series Legendre's differential equation $(1-x^2)\frac{d^2y}{dx^2} - 2x\frac{dy}{dx} + 2y = 0$.

Solve $(D^2+a^2)y = \tan ax$ by using method of variation of parameters.

a) Eliminate the arbitrary function f from the relation $z = y^2 + 2f(\frac{1}{x} + \log y)$.

b) Solve the partial differential equation $\frac{\partial^2 z}{\partial x^2} - \frac{\partial^2 z}{\partial y^2} = x^2y$.

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

b) Find all values of K such that $f(z) = e^x(\cos ky + i\sin ky)$ is analytic.

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

b) If $f(z) = \frac{1}{(z-1)(z-2)^2}$ find residue of all poles.

a) If $\vec{r}$ is the position vector of any point in space, then prove that $r^n\vec{r}$ is irrotational.

b) Find the workdone by the force $\vec{F} = z\hat{i} + x\hat{j} + y\hat{k}$, when it moves a particle along the arc of the curve $\vec{r} = \cos t\hat{i} + \sin t\hat{j} - t\hat{k}$ from $t=0$ to $t=2\pi$.

Verify stokes theorem for $\vec{F}=(x^2-y^2)\hat{i} + 2xy\hat{j}$ over the box bounded by the planes $x=0, x=a, y=0, y=b$.