• Attempt any five questions.
• All questions carry equal marks.
Solve x(dy/dx) + y = x³y⁶ using Bernoulli's
Solve the differential equation (xeʸ+2y)(dy/dx) + y eʸ = 0 using Exact method
Solve (D²-6D+13)y = 8e³ˣ sin 2x
Show that d/dx[xⁿ Jₙ(x)] = xⁿ Jₙ₋₁(x)
Solve (D²+1)y = x sin x using variation of parameters
Form the partial differential equation by eliminating the arbitrary functions from Z=(x+y)φ(x²-y²)
Solve (D²-DD'-6D'²)Z = xy
Solve the partial differential equation yp - xp = z
Show that u = e⁻ˣ(x sin y - y cos y) is Harmonic
Evaluate ∫(3x²+4xy+ix²)dz along y=x² from (0,0) to (1,1)
Find the Poles and Residues at each pole of f(z) = sin²z/(z-π/6)²
Verify Green's theorem in the plane for ∮[(x²-xy³)dx+(y²-2xy)dy] where C is a square with vertices (0,0), (2,0), (2,2), (0,2)
Find the directional derivative of f(x,y,z) = xy²+yz³ at point (2, -1, 1) in the direction of vector î+2ĵ+2k̂
Write short note on: Cauchy's integral formula
Write short note on: Solenoidal and Irrotational
