BT-202 Mathematics - II - Jun 2025 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 $(e^y + 1)\cos x\,dx + e^y\sin x\,dy = 0$.
   How to answer

   • Separate variables: $\frac{\cos x}{\sin x}\,dx + \frac{e^y}{e^y + 1}\,dy = 0$
   • Integrate both sides: $\ln|\sin x| + \ln(e^y + 1) = \ln C$
   • Combine logarithms to write the general solution: $(e^y + 1)\sin x = C$
2. 7 Marks
   bSolve $(D^2 - 5D + 6)y = 4e^x + 5$.
   How to answer

   • Find auxiliary equation: $m^2 - 5m + 6 = 0 \implies m = 2, 3$
   • Write Complementary Function: $y_{CF} = c_1 e^{2x} + c_2 e^{3x}$
   • Evaluate Particular Integral $y_{PI} = \frac{4e^x}{1^2 - 5(1) + 6} + \frac{5}{0 - 0 + 6} = 2e^x + \frac{5}{6}$
   • Write complete solution: $y = y_{CF} + y_{PI} = c_1 e^{2x} + c_2 e^{3x} + 2e^x + \frac{5}{6}$
3. 7 Marks
   aSolve $x^2 \frac{d^2 y}{dx^2} + 5x \frac{dy}{dx} + 4y = x \log x$.
   How to answer

   • Transform by setting $x = e^z$ or $z = \log x$, reducing to $[D(D-1) + 5D + 4]y = z e^z$
   • Simplify operator: $(D^2 + 4D + 4)y = (D+2)^2 y = z e^z$
   • Roots of auxiliary equation: $m = -2, -2 \implies y_{CF} = (c_1 + c_2 z)e^{-2z}$
   • Compute $y_{PI} = \frac{1}{(D+2)^2}(z e^z) = e^z \frac{1}{(D+3)^2} z = \frac{1}{9} e^z (z - \frac{2}{3})$
   • Substitute back $z = \log x$: $y = (c_1 + c_2 \log x)x^{-2} + \frac{x}{9}(\log x - \frac{2}{3})$
4. 7 Marks
   bSolve in series Legendre's differential equation $$(1 - x^2)\frac{d^2 y}{dx^2} - 2x\frac{dy}{dx} + 2y = 0.$$
   How to answer

   • Assume series solution $y = \sum_{m=0}^{\infty} a_m x^m$ around the ordinary point $x=0$
   • Substitute $y, y', y''$ into Legendre equation with $n=1$
   • Obtain recurrence relation: $a_{m+2} = \frac{m-1}{m+1} a_m$
   • Calculate coefficients for odd powers ($a_1 x$, then $a_3 = 0, \dots$) and even powers ($a_2 = -a_0, a_4 = -a_0/3, \dots$)
   • Write general solution as linear combination of two independent series: $y = a_1 x + a_0 [1 - x^2 - \frac{x^4}{3} - \dots]$
5. 14 Marks
   Solve $(D^2 + a^2)y = \tan ax$ by using method of variation of parameters.
   How to answer

   • Find $y_{CF}$: $y_1 = \cos ax$, $y_2 = \sin ax$
   • Calculate Wronskian $W = y_1 y_2' - y_2 y_1' = a(\cos^2 ax + \sin^2 ax) = a$
   • Apply formulas for parameters: $u = -\int \frac{\sin ax \tan ax}{a} dx = -\frac{1}{a^2}[\ln|\sec ax + \tan ax| - \sin ax]$
   • Compute $v = \int \frac{\cos ax \tan ax}{a} dx = -\frac{1}{a^2}\cos ax$
   • Formulate particular integral $y_{PI} = u y_1 + v y_2$ and write complete solution $y = y_{CF} + y_{PI}$
6. 7 Marks
   aEliminate the arbitrary function $f$ from the relation $$z = y^2 + 2f\left(\frac{1}{x} + \log y\right).$$
   How to answer

   • Differentiate relation with respect to $x$: $p = \frac{\partial z}{\partial x} = 2 f'\left(\frac{1}{x} + \log y\right)\left(-\frac{1}{x^2}\right)$
   • Differentiate relation with respect to $y$: $q = \frac{\partial z}{\partial y} = 2y + 2 f'\left(\frac{1}{x} + \log y\right)\left(\frac{1}{y}\right)$
   • Express $2f'$ from the first equation: $2f' = -x^2 p$
   • Substitute into the second equation: $q = 2y + (-x^2 p)\frac{1}{y} \implies y q = 2y^2 - x^2 p$
   • Rearrange to obtain the final PDE: $x^2 p + y q = 2y^2$
7. 7 Marks
   bSolve the partial differential equation $$\frac{\partial^2 z}{\partial x^2} - \frac{\partial^2 z}{\partial y^2} = x^2 y.$$
   How to answer

   • Write in operator form: $(D^2 - D'^2)z = x^2 y$
   • Find CF: Auxiliary equation $m^2 - 1 = 0 \implies m = \pm 1$, so $z_{CF} = \phi_1(y+x) + \phi_2(y-x)$
   • Calculate PI: $z_{PI} = \frac{1}{D^2}\left(1 - \frac{D'^2}{D^2}\right)^{-1}(x^2 y) = \frac{1}{D^2}\left(1 + \frac{D'^2}{D^2}\right)(x^2 y) = \frac{1}{D^2}(x^2 y)$ as $D'^2(y) = 0$
   • Integrate twice with respect to $x$: $\frac{1}{D^2}(x^2 y) = y \frac{x^4}{12}$
   • Write complete solution: $z = \phi_1(y+x) + \phi_2(y-x) + \frac{x^4 y}{12}$
8. 7 Marks
   aSolve $(y+z)p + (x+z)q = x+y$.
   How to answer

   • Write Lagrange auxiliary equations: $\frac{dx}{y+z} = \frac{dy}{x+z} = \frac{dz}{x+y}$
   • Use multipliers $(1, -1, 0)$: $\frac{dx - dy}{-(x-y)} \implies \ln(x-y) + z = \text{const}$ or pairwise subtraction: $\frac{dx-dy}{-(x-y)} = \frac{dy-dz}{-(y-z)} \implies \frac{x-y}{y-z} = c_1$
   • Use multipliers $(1, 1, 1)$: $\frac{dx+dy+dz}{2(x+y+z)} = \frac{dx-dy}{-(x-y)} \implies (x-y)^2(x+y+z) = c_2$ (or similar integral combination)
   • Combine independent integrals: $\phi(c_1, c_2) = 0$
9. 7 Marks
   bFind all values of $K$ such that $f(z) = e^x(\cos ky + i\sin ky)$ is analytic.
   How to answer

   • Identify $u = e^x \cos ky$ and $v = e^x \sin ky$
   • Compute partial derivatives: $u_x = e^x \cos ky$, $u_y = -k e^x \sin ky$, $v_x = e^x \sin ky$, $v_y = k e^x \cos ky$
   • Apply Cauchy-Riemann equations: $u_x = v_y \implies e^x \cos ky = k e^x \cos ky \implies k = 1$
   • Check second C-R condition: $u_y = -v_x \implies -k e^x \sin ky = -e^x \sin ky \implies k = 1$
   • Conclude that $k = 1$
10. 7 Marks
    aEvaluate $\int_{(0,0)}^{(1,1)} (3x^2 + 4xy + ix^2)\,dz$ along $y = x^2$.
    How to answer

    • Use parameterization along $y = x^2$: $dy = 2x\,dx$, so $dz = dx + i\,dy = (1 + 2ix)dx$
    • Substitute $y = x^2$ into the integrand: $3x^2 + 4x(x^2) + ix^2 = 3x^2 + 4x^3 + ix^2$
    • Formulate integral: $\int_0^1 (3x^2 + 4x^3 + ix^2)(1 + 2ix)\,dx$
    • Expand real and imaginary parts of the integrand
    • Integrate with respect to $x$ from $0$ to $1$ to find the numerical complex result
11. 7 Marks
    bIf $f(z) = \frac{1}{(z-1)(z-2)^4}$, find residue of all poles.
    How to answer

    • Identify poles: simple pole at $z=1$ and pole of order 4 at $z=2$
    • Calculate residue at simple pole $z=1$: $\text{Res}(f, 1) = \lim_{z\to 1}(z-1)f(z) = \frac{1}{(1-2)^4} = 1$
    • Calculate residue at $z=2$: $\text{Res}(f, 2) = \frac{1}{3!}\lim_{z\to 2}\frac{d^3}{dz^3}(z-1)^{-1}$
    • Differentiate $(z-1)^{-1}$ three times: $(-1)(-2)(-3)(z-1)^{-4} = -6(z-1)^{-4}$
    • Evaluate at $z=2$: $\text{Res}(f, 2) = \frac{1}{6}(-6)(1)^{-4} = -1$
12. 7 Marks
    aIf $\bar{r}$ is the position vector of any point in space, then prove that $r^n \bar{r}$ is irrotational.
    How to answer

    • State definition of irrotational vector: $\nabla \times \bar{F} = \bar{0}$
    • Use vector identity: $\nabla \times (\phi \bar{A}) = (\nabla \phi) \times \bar{A} + \phi (\nabla \times \bar{A})$
    • Apply to $r^n \bar{r}$: $\nabla \times (r^n \bar{r}) = (\nabla r^n) \times \bar{r} + r^n (\nabla \times \bar{r})$
    • Show $\nabla r^n = n r^{n-2} \bar{r}$ and $\nabla \times \bar{r} = \bar{0}$
    • Compute cross product: $(n r^{n-2} \bar{r}) \times \bar{r} = \bar{0}$ since $\bar{r} \times \bar{r} = \bar{0}$, hence proved
13. 7 Marks
    bFind the workdone by the force $\bar{F} = z\bar{i} + x\bar{j} + y\bar{k}$, when it moves a particle along the arc of the curve $\bar{r} = \cos t\,\bar{i} + \sin t\,\bar{j} - t\bar{k}$ from $t = 0$ to $t = 2\pi$.
    How to answer

    • Express curve coordinates: $x = \cos t, y = \sin t, z = -t$
    • Compute velocity vector: $d\bar{r}/dt = -\sin t\,\bar{i} + \cos t\,\bar{j} - \bar{k}$
    • Express force in terms of $t$: $\bar{F} = -t\bar{i} + \cos t\,\bar{j} + \sin t\,\bar{k}$
    • Calculate dot product: $\bar{F} \cdot d\bar{r}/dt = t\sin t + \cos^2 t - \sin t$
    • Integrate from $t=0$ to $2\pi$: $\int_0^{2\pi} (t\sin t + \cos^2 t - \sin t)\,dt = [-t\cos t + \sin t]_0^{2\pi} + \pi - 0 = -2\pi + \pi = -\pi$
14. 14 Marks
    Verify stokes theorem for $\bar{F} = (x^2 - y^2)\bar{i} + 2xy\bar{j}$ over the box bounded by the planes $x = 0$, $x = a$, $y = 0$, $y = b$.
    How to answer

    • State Stokes theorem: $\oint_C \bar{F} \cdot d\bar{r} = \iint_S (\nabla \times \bar{F}) \cdot \hat{k}\,dx\,dy$
    • Evaluate $\nabla \times \bar{F} = \bar{k}[\frac{\partial}{\partial x}(2xy) - \frac{\partial}{\partial y}(x^2 - y^2)] = \bar{k}(2y - (-2y)) = 4y\bar{k}$
    • Compute surface integral: $\int_0^a dx \int_0^b 4y\,dy = a [2y^2]_0^b = 2ab^2$
    • Evaluate line integral $\oint_C \bar{F} \cdot d\bar{r}$ along four edges of the rectangle $[(0,0)\to(a,0)\to(a,b)\to(0,b)\to(0,0)]$
    • Sum line integrals along all four sides to get $2ab^2$ and verify equality with the surface integral