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Submit Papers đŠ(c) Solve : $\frac{d^2y}{dx^2} - 4y = e^x + \sin 2x$
2. (a) Solve : $x \frac{d^2y}{dx^2} - (2x - 1) \frac{dy}{dx} + (x - 1)y = 0$
(b) Solve by reducing it into its normal form :
$\frac{d^2y}{dx^2} - 2 \tan x \frac{dy}{dx} + y = 0.$
EY-157
B.Tech. IIIrd Semester (New Scheme)
Examination, 2023-24
Engg. Mathematics II
Paper - EL-301
(c) Apply the method of variation of parameters to solve : 6
$\frac{d^2y}{dx^2} + a^2y = \text{cosec } ax$
3. (a) Construct the partial differential equation (P.D.E.) by
eliminating the arbitrary constants.
(i) $z = (x^2 + a)(y^2 + b)$
Note :- Attempt any two parts from each question. All questions
carry equal marks.
(a) Solve : $y (1 + xy) dx + x (1 - xy) dy = 0$
(b) Solve : $y = x + a \tan^{-1} p$
(ii) $2z = \frac{x^2}{a^2} + \frac{y^2}{b^2}$
(b) Solve using Lagrange's method
$xzp + yzq = xy$
(c) Solve : $\frac{\partial^2 z}{\partial x^2} + \frac{\partial^2 z}{\partial y^2} - 6 = \cos (3x + y)$
(a) Show that the function $u(x, y) = e^x \cos y$ is harmonic.
Determine its harmonic conjugate $v(x, y)$ and the analytic
function $f(z) = u + iv.$
(b) Evaluate $\int_0^1 (\bar{z})^2 dz$ along.
(i) The real axis to 2 and then vertically to $2 + i$
(ii) The line $y = \frac{1}{2} x$
(c) Evaluate : $\int_C \frac{(4 - 3z)}{(z - 1)(z - 2)} dz$,
Where C is the circle $|z| = 3/2$
(a) If $\vec{F} = x\vec{i} + y\vec{j} + z\vec{k}$, show $\text{div } \vec{F} = 3$
(ii) $\text{curl } \vec{F} = \vec{0}$
(b) Find the directional derivative of the function $f = x^2 - y^2 + 2x^2$
at the point $P (1, 2, 3)$ in the direction of the line
$PQ$, where $Q$ is the point $(5.04)$.
(c) Evaluate $\int_C \vec{F}.d\vec{r}$, Where $\vec{F} = -xy^2 \vec{i} + y^2 \vec{j} + z\vec{k}$ and $C$
is a circular path $x^2 + y^2 = 4, z = 0$ from $(z, 00)$ to
$(0, 20)$.