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BT-202 · Mathematics II/Unsolved PYQ Paper

BT-202 Mathematics-II - Jun 2024 Question Paper

  1. 7 Marks
    aSolve $x\frac{dy}{dx} + y = x^{3}y^{6}$ using Bernoulli's.
  2. 7 Marks
    bSolve the differential equation $\left(xe^{xy} + 2y\right)\frac{dy}{dx} + ye^{xy} = 0$ using Exact method.
  3. 7 Marks
    aSolve $(D^{2} - 6D + 13)y = 8e^{3x}\sin 2x$.
  4. 7 Marks
    bShow that $\frac{d}{dx}\left[x^{n}J_{n}(x)\right] = x^{n}J_{n-1}(x)$.
  5. 14 Marks
    Solve $(D^{2} + 1)y = x\sin x$ using variation of parameters.
  6. 7 Marks
    aForm the partial differential equation (By eliminating the arbitrary functions) from $Z = (x + y)\phi\left(x^{2} - y^{2}\right)$.
  7. 7 Marks
    bSolve $(D^{2} - DD^{1} - 6D^{12})Z = xy$.
  8. 7 Marks
    aSolve the partial differential equation $yp - xq = z$.
  9. 7 Marks
    bShow that $u = e^{-x}(x\sin y - y\cos y)$ is Harmonic.
  10. 7 Marks
    aEvaluate $\int_{(0,0)}^{(1,1)}\left(3x^{2} + 4xy + ix^{2}\right)dz$ along $y = x^{2}$.
  11. 7 Marks
    bFind the Poles and Residues at each pole of $$f(z) = \frac{\sin^{2} z}{\left(z - \frac{\pi}{6}\right)^{2}}.$$
  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 $(0, 0), (2, 0), (2, 2), (0, 2)$.
  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 $\bar{i} + 2\bar{j} + 2\bar{k}$.
  14. 7 Marks
    bWrite short note on: i) Cauchy's integral formula ii) Solenoidal and Irrotational
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