Mathematics II (BT-202) - Important Questions

Solve (1+y^2)dx = (tan^{-1}y - x)dy as a linear equation in x as function of y.

Solve second-order linear differential equation with constant coefficients (D^2 - 4D + 3)y = cos 2x.

Solve x^2 d^2y/dx^2 + 5x dy/dx + 4y = x log x, a Cauchy-Euler homogeneous linear differential equation, by reduction to constant coefficients.

Show that J_{1/2}(x) = sqrt(2/(pi x)) sin x.

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

Solve the first-order linear partial differential equation (y+z)p + (x+z)q = x+y by Lagrange's method.

Solve (D^2 + 4DD' - 5D'^2)Z = sin(2x + 3y), a homogeneous linear partial differential equation with constant coefficients with sinusoidal right-hand side.

Find all values of K such that f(z) = e^x(cos Ky + i sin Ky) is analytic using Cauchy-Riemann equations.

Show that u(x,y) = e^x cos y is harmonic and find its harmonic conjugate v(x,y) and construct the corresponding analytic function f(z) = u + iv.

Using Cauchy integral formula for derivatives, evaluate integral_C z^3 e^{-z}/(z-1)^3 dz where C is |z-1| = 1/2.

Find the poles and the residues at each pole for f(z) = 1/((z-1)(z-2)^4) involving simple and higher-order poles.

If bar{r} is the position vector of any point in space, prove that r^n bar{r} is irrotational, i.e. curl(r^n bar{r}) = bar{0}.

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

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