BT-202 (GS) – Mathematics-II
• Attempt any five questions.
• All questions carry equal marks.
a)Unit 1Solve dy/dx = cos(x+y) + sin(x+y)
Solve dy/dx = cos(x+y) + sin(x+y)
b)Unit 1Solve (1+y²)dx = (tan⁻¹y - x)dy
Solve (1+y²)dx = (tan⁻¹y - x)dy
a)Unit 1Solve d²y/dx² + dy/dx = (1+eˣ)⁻¹
Solve d²y/dx² + dy/dx = (1+eˣ)⁻¹
b)Unit 1Solve dx/dt - y = eᵗ, dy/dt + x = sin t with initial conditions x(0)=1, y(0)=0
Solve dx/dt - y = eᵗ, dy/dt + x = sin t with initial conditions x(0)=1, y(0)=0
Unit 2Solve the differential equation x(1-x)y'' + 2(1-2x)y' - 2y = 0 using Frobenius method
Solve the differential equation x(1-x)y'' + 2(1-2x)y' - 2y = 0 using Frobenius method
a)Unit 2Prove that J₁/₂(x) = √(2/πx) sin x
Prove that J₁/₂(x) = √(2/πx) sin x
b)Unit 3Solve by Charpit's method, the PDE (p²+q²)y = qz
Solve by Charpit's method, the PDE (p²+q²)y = qz
a)Unit 3Solve (D² - 6DD' + 9D'²)z = 12x² + 36xy
Solve (D² - 6DD' + 9D'²)z = 12x² + 36xy
b)Unit 4Prove that an analytic function with constant modulus is constant
Prove that an analytic function with constant modulus is constant
a)Unit 4Use Cauchy Integral formula to solve ∮[sin πz² + cos πz²]/[(z-1)(z-2)] dz where C is circle |z|=3
Use Cauchy Integral formula to solve ∮[sin πz² + cos πz²]/[(z-1)(z-2)] dz where C is circle |z|=3
b)Unit 4Using complex integration method, solve ∫₀²ᵖⁱ cos4θ/(5+4cosθ) dθ
Using complex integration method, solve ∫₀²ᵖⁱ cos4θ/(5+4cosθ) dθ
a)Unit 4Solve ∫₀¹⁺ⁱ (x-y+ix²)dz along real axis from z=0 to z=1 and then along line parallel to imaginary axis from z=1 to z=1+i
Solve ∫₀¹⁺ⁱ (x-y+ix²)dz along real axis from z=0 to z=1 and then along line parallel to imaginary axis from z=1 to z=1+i
b)Unit 5Prove that ∇²f(r) = f''(r) + (2/r)f'(r)
Prove that ∇²f(r) = f''(r) + (2/r)f'(r)
a)Unit 5Find the directional derivative of f(x,y,z) = e²ˣ cos yz at (0,0,0) in direction of tangent to curve x=a sin t, y=a cos t, z=at at t=π/4
Find the directional derivative of f(x,y,z) = e²ˣ cos yz at (0,0,0) in direction of tangent to curve x=a sin t, y=a cos t, z=at at t=π/4
b)Unit 5Using Green's theorem, find the area of the region in first quadrant bounded by curves y=x, y=1/x, y=x/4
Using Green's theorem, find the area of the region in first quadrant bounded by curves y=x, y=1/x, y=x/4