Solve the following differential equations:

1 (a) (i)
\[ xy \dfrac {dy}{dx} = (1-x^2) (1+y^2) \]

4 M

1 (a) (ii)
\[ \cos y - x \sin y \dfrac {dy}{dx} = \sec^2 x. \]

4 M

1 (b)
An e.m.f. 200e

^{-5t}is applied to a series circuit containing of 20 ohm resistor and 0.01 F capacitor. Find the charge and current at any time assuming that there is no initial charge on the capacitor.
4 M

2 (a)
Solve: 2y dx+(2 x log x ? xy)dy=0.

4 M

2 (b) (i)
A body starts moving from rest is opposed by a force per unit mass of value cx and resistance per unit mass of value bv

^{2}, where x and v are the displacement and velocity of the body at that instant. Show that the velocity of the body is given by: \[ v^2 = \dfrac {c}{2b^2} (1-e^{-2bx}) - \dfrac {cx}{b} \]
4 M

2 (b) (ii)
The inner and outer surface of a spherical shell are maintained at T

_{0}and T_{1}temperature respectively. If the inner and outer radii of the shell are r_{0}and r_{1}respectively and thermal conductivity of the shell is k, find the amount of heat loss from the shell per unit time. Find also the temperature distribution through the shell.
4 M

3 (a)
Obtain the first three coefficient in the Fourier cosine series for y using practical harmonic analysis:

x |
y |

0 | 4 |

1 | 8 |

2 | 15 |

3 | 7 |

4 | 6 |

5 | 2 |

5 M

3 (b)
Evaluate: \( \int^5_3 (x-3)^{1/2} (5-x)^{1/2} dx. \)

3 M

Trace the following curve (any one):

3 (c) (i)
ay

^{2}=x^{2}(a-x)
4 M

3 (c) (ii)
r=a(1 + \cos \theta).

4 M

4 (a)
\[ \text{If } I_n=\int^{\pi / 2}{0} x^n \cos x \ dx \] Prove that: \[ I_n = \left ( \dfrac {\pi}{2} \right )^n - n (n-1) I_{n-2} \]

4 M

4 (b)
Show that: \[ \int^{\infty}_0 e^{-x^{2}-2bx} dx = \dfrac {\sqrt{\pi}}{2} e^{b^2} [1-erf (b)] \]

4 M

4 (c)
Find the arc length of the curve (using rectification) r=2a cos θ.

4 M

5 (a)
Find the equation of the sphere which passes through the point (3, 1, 2) and meets X to Y plane in a circle of radius 3 units with centre at (1, -2, 0).

5 M

5 (b)
Find the equation of right circular cone whose vertex is at the origin with axis \( \dfrac {x}{1} = \dfrac {y}{2} = \dfrac {z}{3} \) and has a semi-vertical angle 30°.

4 M

5 (c)
Find the equation of right circular cylinder of radius 2 whose axis passes through (1, 2, 3) and has direction cosines proportional to 2, 1, 2.

4 M

6 (a)
Find the equation of the sphere passing through the circle x

^{2}+y^{2}+z^{2}=9, 2x+3y+4z=5 and the point (1, 2, 3).
5 M

6 (b)
Find the equation of right circular cone whose vertex is (1, -1, 1) and axis is parallel to \( x=\dfrac {-y}{2} =-z\) and one of its generators has direction cosines proportional to (2, 2, 1).

4 M

6 (c)
Find the equation of right circular cylinder of radius 4 with axis passing through origin and making equal angles with the co-ordinate axes.

4 M

Attempt any two of the following:

7 (a)
\[ \text{Evaluate :} \int^{1}_0 dx \int^{\infty}_{1} e^{-y} y^x \log y \ dy. \]

6 M

7 (b)
Evaluate :\[ \iiint (x^2y^2 + y^2z^2+z^2x^2) dx \ dy \ dz. \] throughout the volume of the sphere x

^{2}+y^{2}+z^{2}=a^{2}.
7 M

7 (c)
Find the moment of inertia of one loop of the lemniscate r

^{2}=a^{2}cos 2θ about initial line.
6 M

8 (a)
Evaluate: \[ \int^a_0 \int^{\sqrt{a^2 - x^2}}_0 \sin \left \{ \dfrac {\pi}{a^2} (a^2 - x^2 - y^2) \right \}dx \ dy. \]

7 M

8 (b)
Evaluate: \[ \int^\infty_0 \int^\infty_0 \int^\infty_0 \dfrac {dx \ dy \ dz }{(1+x^2 + y^2 + z^2)^2}. \]

6 M

8 (c)
Find the C.G. of the loop of the curve:

y

y

^{2}(a+x)=x^{2}(a-x).
6 M

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