BT-202 Mathematics-II - Jun 2024 Question Paper

Under each question, How to answer lists the points an answer should cover, outlined by CampusPrep from the syllabus. Check the details against your textbook.

1. 7 Marks
   aSolve $x\frac{dy}{dx} + y = x^{3}y^{6}$ using Bernoulli's.
   How to answer

   • Rewrite as Bernoulli form dy/dx + (1/x)y = x^2 y^6
   • Substitute v = y^{-5} to linearize
   • Find integrating factor and integrate
   • Back-substitute to get general solution
2. 7 Marks
   bSolve the differential equation $\left(xe^{xy} + 2y\right)\frac{dy}{dx} + ye^{xy} = 0$ using Exact method.
   How to answer

   • Write in form M dx + N dy = 0 and check exactness
   • Verify ∂M/∂y = ∂N/∂x
   • Integrate to find potential function
   • Write implicit general solution
3. 7 Marks
   aSolve $(D^{2} - 6D + 13)y = 8e^{3x}\sin 2x$.
   How to answer

   • Find complementary function from auxiliary equation D^2-6D+13=0
   • Write particular integral for 8e^{3x}sin2x
   • Evaluate 1/f(D) using exponential-shift / inverse operator
   • Combine C.F. + P.I. as general solution
4. 7 Marks
   bShow that $\frac{d}{dx}\left[x^{n}J_{n}(x)\right] = x^{n}J_{n-1}(x)$.
   How to answer

   • Start from series definition of Bessel function J_n(x)
   • Differentiate x^n J_n(x) termwise
   • Use recurrence / index shift to simplify
   • Obtain x^n J_{n-1}(x)
5. 14 Marks
   Solve $(D^{2} + 1)y = x\sin x$ using variation of parameters.
   How to answer

   • Find C.F. from D^2+1=0 as C1 cos x + C2 sin x
   • Compute Wronskian and variation-of-parameters formulas
   • Integrate to find u1(x), u2(x) for RHS x sin x
   • Write complete solution y = C.F. + P.I.
6. 7 Marks
   aForm the partial differential equation (By eliminating the arbitrary functions) from $Z = (x + y)\phi\left(x^{2} - y^{2}\right)$.
   How to answer

   • Compute p = ∂Z/∂x and q = ∂Z/∂y with arbitrary φ
   • Eliminate φ and φ' between Z, p, q equations
   • Simplify to first-order PDE in p, q, x, y, Z
   • Write final PDE
7. 7 Marks
   bSolve $(D^{2} - DD^{1} - 6D^{12})Z = xy$.
   How to answer

   • Write auxiliary equation in m = D/D1 and solve for m
   • Write complementary function with arbitrary functions of x+my
   • Find particular integral for RHS xy by inverse operator
   • Combine C.F. + P.I.
8. 7 Marks
   aSolve the partial differential equation $yp - xq = z$.
   How to answer

   • Identify Lagrange form Pp+Qq=R with P=y, Q=-x, R=z
   • Solve auxiliary equations dx/y = dy/(-x) = dz/z
   • Find two independent integrals u=c1, v=c2
   • Write general solution F(u,v)=0
9. 7 Marks
   bShow that $u = e^{-x}(x\sin y - y\cos y)$ is Harmonic.
   How to answer

   • Recall harmonic means ∇²u = 0
   • Compute first and second partial derivatives of u
   • Add u_xx + u_yy (+ u_zz if considered)
   • Show sum is zero
10. 7 Marks
    aEvaluate $\int_{(0,0)}^{(1,1)}\left(3x^{2} + 4xy + ix^{2}\right)dz$ along $y = x^{2}$.
    How to answer

    • Parametrize path y=x², z=x+iy, dz=(1+i2x)dx
    • Substitute into integrand (3x²+4xy+ix²)
    • Integrate x from 0 to 1 (real + imaginary parts)
    • Evaluate to complex value
11. 7 Marks
    bFind the Poles and Residues at each pole of $$f(z) = \frac{\sin^{2} z}{\left(z - \frac{\pi}{6}\right)^{2}}.$$
    How to answer

    • Identify singularity at z=π/6 from denominator
    • Determine order of pole (order 2)
    • Apply residue formula for multiple pole
    • State pole and residue value
12. 14 Marks
    Verify Green's theorem in the plane for $$\int_{C}\left(x^{2} - xy^{3}\right)dx + \left(y^{2} - 2xy\right)dy$$ where C is a square with vertices $(0, 0), (2, 0), (2, 2), (0, 2)$.
    How to answer

    • State Green's theorem ∮_C Pdx+Qdy = ∬_R (∂Q/∂x-∂P/∂y)dA
    • Evaluate line integral along four sides of square
    • Evaluate double integral over 0≤x,y≤2
    • Show both sides equal
13. 7 Marks
    aFind the directional derivative of $f(x, y, z) = xy^{2} + yz^{3}$ at point $(2, -1, 1)$ in the direction of the vector $\bar{i} + 2\bar{j} + 2\bar{k}$.
    How to answer

    • Compute gradient ∇f at (2,-1,1)
    • Find unit vector in direction i+2j+2k
    • Take dot product ∇f·unit vector
    • Evaluate numeric value
14. 7 Marks
    bWrite short note on: i) Cauchy's integral formula ii) Solenoidal and Irrotational
    How to answer

    • State Cauchy's integral formula with conditions
    • Give proof idea / applications
    • Define solenoidal (div=0) with example
    • Define irrotational (curl=0) with example