• Attempt any five questions.
• All questions carry equal marks.
Solve (1+y²)dx = (tan⁻¹y - x)dy
Solve the simultaneous differential equations: d²x/dt² - 4dx/dt + 4x = y and d²y/dt² + 4dy/dt + 4y = 25x + 16eᵗ
Find the series solution of xy''+y'+xy=0 about x=0 by the Frobenius method
Apply the method of variation of parameters to solve d²y/dx² + y = tan x
Solve (D²-2DD'+D'²)Z = eˣ⁺²ʸ + x²
Find a complete integral of px + qy = pq
Show that u = eˣ(x cos y - y sin y) is a harmonic function. Find the conjugate function v
Evaluate ∮ eᶻ/[(z-1)(z-4)] dz where C is circle |z|=2 by using Cauchy's Integral Formula
Find the directional derivative of f(x,y,z) = 2x²+3y²+z² at point P(2,1,3) in direction of vector a = î - 2k̂
Evaluate ∫ F·dr, where F = x²î + y³ĵ and curve C is arc of parabola y=x² in xy plane from (0,0) to (1,1)
Evaluate using residue theorem ∮ (4-3z)/[(z-1)(z-2)z] dz where C is circle |z| = 3/2
Evaluate ∫₀²ᵖⁱ dθ/(5-3cosθ)
Verify Stokes' theorem for vector field F = (x²-y²)î + 2xyĵ integrated around rectangle z=0 bounded by x=0, y=0, x=a and y=b
Define Gradient, Divergence and Curl
Solve (D²-D'²-3D+3D')Z = eˣ⁺²ʸ + xy
By reducing to homogeneous, solve: (1+x)²(d²y/dx²)+(1+x)(dy/dx)+y = 4 cos{log(1+x)}
