Solve any Two of the following:

1 (a) (i)
(D

^{2}+9)y=x.sin 2x
4 M

Answer any one question from Q1 and Q2

1 (a) (ii)
(D

^{2}-6D +13) y=e^{3x}. Sin 4x + 3^{x}.
4 M

1 (a) (iii)
\[ (D^2 + 4)y = \dfrac {1}{1 + \sin 2x}, \] (using method of variation of parameters.)

4 M

1 (b)
Using Fourier cosine integral of e

^{-mx}(m>0) show that: \[ \int\limits_0^{\infty} \dfrac {m \cos \lambda x \ d \lambda}{m^2 + \lambda^2} = \dfrac {\pi}{2}\cdot e^{-mx}, \ (m>0, \ x>0) \]
4 M

2 (a)
A body of weight 'W = 20 N' is hung from a spring. A pull
of '40 N' will stretch the spring to 10 cm. The body is pulled
down to 20 cm below the static equilibrium position and then released. Find the displacement of the body from its equilibrium position in time 't' seconds.

4 M

Solve any one of the following:

2 (b) (i)
Find the Laplace transform of \[ f(t) = \int_\limits^t_0 \dfrac {\sin t} {t} dt \]

4 M

2 (b) (ii)
Find the inverse Laplace transform of \[ F(s) = \log \left ( \dfrac {s+4}{s+8} \right ) \]

4 M

2 (c)
Solve the following differential equation by using Laplace transform method:

y''(t)+9y(t)=18t, Given that: \[ y(0)=0, \ y\left ( \dfrac {\pi}{2} \right )=1 \]

y''(t)+9y(t)=18t, Given that: \[ y(0)=0, \ y\left ( \dfrac {\pi}{2} \right )=1 \]

4 M

Answer any one question from Q3 and Q4

3 (a)
Calculate the first four moments about the mean of the given distribution:

x |
f |

2.0 | 4 |

2.5 | 36 |

3.0 | 60 |

3.5 | 90 |

4.0 | 70 |

4.5 | 40 |

5 | 10 |

4 M

3 (b)
Between 2 p.m. and 3 p.m. the average number of phone calls per minute coming into company are 3. Find the probability that during one particular minute there will be 2 or less calls

4 M

3 (c)
Find the directional derivative of ϕ=xy

^{2}+yz^{3}at (1,-1,1) in the direction of tangent to the curve x=sin t, y=cos t, z=t at t=π/4.
4 M

4 (a)
Find the coefficient of correlation for the following data:

x |
f |

2.0 | 4 |

2.5 | 36 |

3.0 | 60 |

3.5 | 90 |

4.0 | 70 |

4.5 | 40 |

5 | 10 |

4 M

4 (b)
Prove that (any one): \[ i) \ \ \nabla \left ( \dfrac {\overline a \cdot \overline r}{r^n} \right ) = \dfrac {\overline a}{r^n} - \dfrac {n (\overline a \cdot \overline r)}{r^{n+2}}\overline r \\ ii) \ \nabla^4 (r^2 \log r) = \dfrac {6}{r^2}. \]

4 M

4 (c)
Show that \[ F=(6xy+z

^{3})i+(3x^{2}-z)j+(3xz^{2}-y)k is irrotational. Find the scalar ϕ such that: F=∇ϕ.
4 M

Answer any one question from Q5 and Q6

5 (a)
Evaluate: \[ \int_c \overline F\cdot d\overline r \ for \ \\ \overline F=3x^2 \overline i + (2xz - y) \overline j + z\overline k \] along the curve:

x=2t

x=2t

^{2}, y=t, z=4t^{2}-t from t=0 to t=1.
4 M

5 (b)
Evaluate: \[ \iint_s \overline F \cdot d\overline s \ where \\ \overline F = 4xz\overline i + xyz^2 \overline j + 3z\overline k \] and S is the surface of the cone: z

^{2}=x^{2}+y^{2}bounded by z=4.
5 M

5 (c)
Apply Stoke's theorem to calculate: \[ \int_c 4 ydx + 2zdy + 6ydz \] where C is the curve of intersection of x

^{2}+y^{2}+z^{2}=6z and z=x+3.
4 M

6 (a)
Evaluate: \[ \int_c \overline F\cdot d\overline r \ for \\ \overline F = x^2\overline i + 2xy\overline j + z\overline k \] and C is the straight line joining (1, 0, 2), (3, 1, 1).

4 M

6 (b)
Show that: \[ \iint_s \dfrac {\overline r} {r^3} \cdot \widehat n \ ds =0 \]

4 M

6 (c)
Evaluate using Stoke's theorem: \[ \int_c ( ydx + zdy + xdz) \] C being intersection of x

^{2}+y^{2}+z^{2}=a^{2}, x+z=a.
5 M

Answer any one question from Q7 and Q8

7 (a)
A string is stretched and fastened to two points l apart. Motion is started by displacing the string into the form y=k(lx-x

^{2}) from which it is released at time t=0. Find the displacement of any point on the string at a distance of x from one end at time t. Using differential equation: \[ \dfrac {\partial ^ 2y}{\partial t^2} = C^2 \dfrac {\partial ^2 y}{\partial x^2} \]
7 M

7 (b)
The temperature at any point of the insulated metal rod of one metre length is governed by the differential equation: \[ \dfrac {\partial u}{\partial t^2}= C^2 \dfrac {\partial ^2 u}{\partial t^2} \] Find u(x,t) subject to the following:

i) u(0, t) =0°C

ii) u(1, t)=0°C

iii) u(x, 0)=50°C.

i) u(0, t) =0°C

ii) u(1, t)=0°C

iii) u(x, 0)=50°C.

6 M

8 (a)
Solve the partial differential equation: \[ \dfrac {\partila ^2 u}{\partial x^2} + \dfrac {\partial ^2 u}{\partial y^2} =0 \] subject to boundary conditions:

i) u(0, y)=0

ii) u(x, 0)=0

iii) u(a, y)=0

iv) u(x, b)=40.

i) u(0, y)=0

ii) u(x, 0)=0

iii) u(a, y)=0

iv) u(x, b)=40.

7 M

8 (b)
Use Fourier transform to solve the equation: \[ \dfrac {\partial u}{\partial t} = \dfrac {\partial ^2}{\partial x^2}, \ 0< x< \infty, \ t>0 \] subject to the following conditions: \[ i) \ u(0, t)=0, \ t>0 \\ ii) \ u(x,0)= \left\{\begin{matrix}
1 &0< x<1 \\0
&x>1
\end{matrix}\right. \\ iii) \ u(x, t ) \ is \ bounded. \]

6 M

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