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

b) Solve the simultaneous differential equations: $$\frac{d^2x}{dt^2}-4\frac{dx}{dt}+4x=y \quad \text{and} \quad \frac{d^2y}{dt^2}+4\frac{dy}{dt}+4y=25x+16e^t$$ (Unit 1)

a) Find the series solution of $xy''+y'+xy=0$ about $x=0$ by the Frobenius method. (Unit 2)

b) Apply the method of variation of parameters to solve the ordinary differential equation: $\frac{d^2y}{dx^2}+y=\tan x$. (Unit 1)

a) Solve $(D^2-2DD'+D'^2)Z=e^{x+2y}+x^2$. (Unit 3)

b) Find a complete integral of $px+qy=pq$. (Unit 3)

a) Show that the function $u=e^x(x\cos y-y\sin y)$ is a harmonic function. Find the conjugate function $v$. (Unit 4)

b) Evaluate: $\oint_C \frac{e^z}{(z-1)(z-4)}dz$ where C is the circle $|z|=2$ by using Cauchy's Integral Formula. (Unit 4)

a) Find the directional derivative of $f(x,y,z)=2x^2+3y^2+z^2$ at the point $P(2,1,3)$ in the direction of the vector $\vec{a}=\hat{i}-2\hat{k}$. (Unit 5)

b) Evaluate $\int_C \vec{F}\cdot d\vec{r}$, where $\vec{F}=x^2\hat{i}+y^3\hat{j}$ and the curve C is the arc of the parabola $y=x^2$ in the xy plane from $(0,0)$ to $(1,1)$. (Unit 5)

a) Evaluate the following integral using residue theorem $\oint_C \frac{4-3z}{(z-1)(z-2)z}dz$ where C is the circle $|z|=\frac{3}{2}$. (Unit 4)

b) Evaluate the integral $\int_0^{2\pi} \frac{d\theta}{5-3\cos\theta}$. (Unit 4)

a) Verify Stokes' theorem for the vector field $\hat{F}=(x^2-y^2)\hat{i}+2xy\hat{j}$ integrated around the rectangle $z=0$ and bounded by the line $x=0, y=0, x=a$ and $y=b$. (Unit 5)

b) Define Gradient, Divergence and Curl. (Unit 5)

a) Solve $(D^2-D'^2-3D+3D')Z=e^{x+2y}+xy$. (Unit 3)

b) By reducing to homogeneous, solve the differential equation: $$(1+x)^2\frac{d^2y}{dx^2}+(1+x)\frac{dy}{dx}+y=4\cos\{\log(1+x)\}$$ (Unit 1)