BT-202 Mathematics - II - Dec 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 $(1+y^2)dx = (\tan^{-1}y - x)dy$.
   How to answer

   • Rewrite as linear in x vs y: dx/dy + x/(1+y^2) = tan^{-1}y/(1+y^2)
   • Integrating factor IF = exp(arctan y)
   • Solution x*IF = integral tan^{-1}y/(1+y^2)*IF dy + C
2. 7 Marks
   bSolve $(D^2+3D+2)y = \sin 3x$.
   How to answer

   • Auxiliary equation m^2+3m+2=0, roots m=-1,-2 for C.F.
   • Particular integral (D^2+3D+2)^{-1} sin3x
   • Replace D^2 by -9 and simplify to a sin3x + b cos3x form
3. 7 Marks
   aSolve the simultaneous equations $\frac{dx}{dt}-7x+y=0$ and $\frac{dy}{dt}-2x-5y=0$.
   How to answer

   • Write in operator D = d/dt form: (D-7)x + y = 0, -2x + (D-5)y = 0
   • Eliminate y to get second order equation in x
   • Solve for x(t) then obtain y(t) from first equation
4. 7 Marks
   bSolve by the method of variation of parameter $(D^2+1)y = x$.
   How to answer

   • C.F. y_c = C1 cos x + C2 sin x
   • Wronskian and variation of parameters formulas for u1, u2
   • Particular integral and complete solution y = y_c + y_p
5. 7 Marks
   aSolve $(1+x)^2\frac{d^2y}{dx^2}+(1+x)\frac{dy}{dx}+y = \cos\log(1+x)$.
   How to answer

   • Identify Cauchy-Euler type with 1+x; put 1+x = e^z
   • Transformed constant coefficient equation (D(D-1)+D+1)y = cos z
   • Solve for C.F. and P.I. and back-substitute z = log(1+x)
6. 7 Marks
   bShow that $J_n(-x) = (-1)^n J_n(x)$ when $n$ is positive or negative integer.
   How to answer

   • Start from series definition of J_n(x)
   • Replace x by -x and factor (-1)^{r} terms
   • Show factor reduces to (-1)^n for integer n
7. 7 Marks
   aSolve by Charpit's method $px+qy = pq$.
   How to answer

   • Identify Charpit's auxiliary equations for px+qy=pq
   • Find first integral u involving p,q
   • Obtain complete integral with two arbitrary constants
8. 7 Marks
   bSolve the Partial differential equation $\left(D^3-4D^2D'+4DD'^{2}\right)Z = \cos(2x+y)$.
   How to answer

   • Write auxiliary equation and find C.F. from (D-2D')^2=0
   • Particular integral 1/(D^3-4D^2D'+4DD'^2) cos(2x+y)
   • Replace D by 2, D' by 1 with failure case handling
9. 7 Marks
   aConstruct a partial differential equation from the relation $f\left(x^2+y^2+z^2, z^2-2xy\right) = 0$
   How to answer

   • Let u = x^2+y^2+z^2, v = z^2-2xy and f(u,v)=0
   • Differentiate partially w.r.t x and y using p,q
   • Eliminate arbitrary function f to get first order PDE
10. 7 Marks
    bShow that $u = e^{-x}(x\sin y - y\cos y)$ is Harmonic.
    How to answer

    • Compute u_x, u_xx and u_y, u_yy
    • Show u_xx + u_yy = 0
    • Conclude u satisfies Laplace equation hence harmonic
11. 7 Marks
    aDetermine P such that the function $f(z) = \frac{1}{2}\log(x^2+y^2) + i\tan^{-1}\left(\frac{px}{y}\right)$ be an analytic function.
    How to answer

    • Identify u = (1/2)log(x^2+y^2), v = tan^{-1}(Px/y)
    • Apply Cauchy-Riemann equations u_x = v_y, u_y = -v_x
    • Solve for P, get P = 1
12. 7 Marks
    bEvaluate using Cauchy's theorem $\int_{c}\frac{z^3 e^{-z}}{(z-1)^3}dz$ where $c$ is $|z-1| = \frac{1}{2}$.
    How to answer

    • Identify singularity z=1 of order 3 inside |z-1|=1/2
    • Apply Cauchy integral formula for derivatives
    • Evaluate 2nd derivative of z^3 e^{-z} at z=1
13. 7 Marks
    aFind the poles and residues at each pole of $\frac{e^{iz}}{z^2+1}$.
    How to answer

    • Factor denominator z^2+1 = (z-i)(z+i) for simple poles
    • Residue at z=i and z=-i via limit formula
    • State poles z=±i with residues e^{-1}/2i and e^{1}/-2i
14. 7 Marks
    bFind the directional derivative of $\varnothing = x^2yz + 4xz^2$ at $(1,-2,-1)$ in the direction of $2\bar{i}-\bar{j}-2\bar{k}$.
    How to answer

    • Compute gradient ∇Ø
    • Unit vector along 2i-j-2k
    • Directional derivative = ∇Ø · unit vector at (1,-2,-1)
15. 14 Marks
    Verify Green's theorem for $\int_{C}\left[(xy+y^2)dx + x^2dy\right]$ where C is the boundary by $y = x$ and $y = x^2$.
    How to answer

    • State Green's theorem ∮_C Mdx+Ndy = ∬_R (N_x - M_y) dxdy
    • Evaluate line integral along y=x and y=x^2
    • Evaluate double integral over region and show equality