1. Solve the following differential equations (5×4 = 20)
(a) xdy ydx x y dx 2 2 − = −
(b) ) 0
1
) (3
1
(4 3 3 4 2 + + − dy =
y
dx x y
x
x y
(c) y x x
dx
d y
+ = + 2
2
2
(d) y x x
dx
dy
dx
d y
4 4 4cos 3 sin
2
2
+ + = +
2. A mass of 60 kg slides on a table. The frictional force is 60 times the velocity and the mass is
being pushed with a force of 54 sin 2t N. Find the velocity as a function of time if v = 0 at
t = 0. (10)
3. A circuit consists of an inductance of 0.05 H, a resistance of 20W, a capacitor of 100 μF and
an emf of 100 cos 200 t. Find i and q given that q = 0 and i = 0 at t = 0. (10)
4. Obtain the power series solution for the following differential equation:
(1 ) 2 2 0
2
2
2 − − + y =
dx
dy
x
dx
d y
x (10)
5. Show that the function (10×2 = 20)
a) u = x4 − 6x2 y2 + y is a solution of the two-dimensional Laplace equation.
b) u = sin w ct sin wx is a solution of the one-dimensional wave equation.
6. Write down the equation
u t c u 2 2 2 2 ¶ / ¶ = Ñ
in plane polar coordinates. By the method of separation of variables, reduce the PDE so
obtained to a set of ODEs. (10)
7. Represent the following function by a Fourier sine series
/ 8 / 4
0 / 8
if
4
if
( )
p < < p
< < p
−
= p
t
t
t
t
f t (10)
8. Find the solution of the wave equation for a vibrating string of length l = 10 units with ends
fixed. It is given that c2 = 1, initial velocity is zero and initial deflection is
u (x, 0) = k (sin x − sin 2x)
1. Solve the following differential equations (5×4 = 20)
(a) xdy ydx x y dx 2 2 − = −
(b) ) 0
1
) (3
1
(4 3 3 4 2 + + − dy =
y
dx x y
x
x y
(c) y x x
dx
d y
+ = + 2
2
2
(d) y x x
dx
dy
dx
d y
4 4 4cos 3 sin
2
2
+ + = +
2. A mass of 60 kg slides on a table. The frictional force is 60 times the velocity and the mass is
being pushed with a force of 54 sin 2t N. Find the velocity as a function of time if v = 0 at
t = 0. (10)
3. A circuit consists of an inductance of 0.05 H, a resistance of 20W, a capacitor of 100 μF and
an emf of 100 cos 200 t. Find i and q given that q = 0 and i = 0 at t = 0. (10)
4. Obtain the power series solution for the following differential equation:
(1 ) 2 2 0
2
2
2 − − + y =
dx
dy
x
dx
d y
x (10)
5. Show that the function (10×2 = 20)
a) u = x4 − 6x2 y2 + y is a solution of the two-dimensional Laplace equation.
b) u = sin w ct sin wx is a solution of the one-dimensional wave equation.
6. Write down the equation
u t c u 2 2 2 2 ¶ / ¶ = Ñ
in plane polar coordinates. By the method of separation of variables, reduce the PDE so
obtained to a set of ODEs. (10)
7. Represent the following function by a Fourier sine series
/ 8 / 4
0 / 8
if
4
if
( )
p < < p
< < p
−
= p
t
t
t
t
f t (10)
8. Find the solution of the wave equation for a vibrating string of length l = 10 units with ends
fixed. It is given that c2 = 1, initial velocity is zero and initial deflection is
u (x, 0) = k (sin x − sin 2x)
