a) Solve: $\frac{dy}{dx} = \cos(x+y) + \sin(x+y)$.

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

a) Solve: $\frac{d^2y}{dx^2} + \frac{dy}{dx} = (1+e^x)^{-1}$.

b) Solve: $\frac{dx}{dt} - y = e^t$, $\frac{dy}{dt} + x = \sin t$; with initial conditions $x(0)=1, y(0)=0$.

Solve the differential equation $x(1-x)y'' + 2(1-2x)y' - 2y = 0$ using Frobenius method.

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

b) Solve by Charpit's method, the P.D.E $(p^2+q^2)y = qz$.

a) Solve: $(D^2 - 6DD' + 9D'^2)z = 12x^2 + 36xy$.

b) Prove that an analytic function with constant modulus is constant.

a) Use Cauchy Integral formula to solve $\oint_C \frac{\sin \pi z^2 + \cos \pi z^2}{(z-1)(z-2)}dz$ where C is the circle $|z|=3$.

b) Using complex integration method, solve: $\int_0^{2\pi} \frac{\cos 4\theta}{5+4\cos\theta}d\theta$.

a) Solve: $\int_0^{1+i} (x-y+ix^2)dz$ along the real axis from $z=0$ to $z=1$ and then along a line parallel to imaginary axis from $z=1$ to $z=1+i$.

b) Prove that: $\nabla^2 f(r) = f''(r) + \frac{2}{r}f'(r)$.

a) Find the directional derivative of $f(x,y,z) = e^{2x} \cos yz$ at $(0,0,0)$ in the direction of the tangent to the curve $x=a \sin t, y=a \cos t, z=at$ at $t=\frac{\pi}{4}$.

b) Using Green's theorem, find the area of the region in the first quadrant bounded by the curve $y=x, y=\frac{1}{x}, y=\frac{x}{4}$.