Mathematical Methods for Physicists: A concise introduction - Site Map

PROBLEMS
Figure 2.3.
RC circuit.
2.4 RC circuits: A typical RC circuit is shown in Fig. 2.3. Find current ¯ow I…t†
in the circuit, assuming E…t† ˆE 0 .
Hint: the voltage drop across the capacitor is given Q/C, with Q(t) the
charge on the capacitor at time t.
2.5 Find a constant such that …x ‡ y† is an integrating factor of the equation
…4x 2 ‡ 2xy ‡ 6y†dx ‡…2x 2 ‡ 9y ‡ 3x†dy ˆ 0:
What is the solution of this equation?
2.6 Solve dy=dx ‡ y ˆ y 3 x:
2.7 Solve:
(a) the equation …D 2 D 12†y ˆ 0 with the boundary conditions y ˆ 0,
Dy ˆ 3 when t ˆ 0;
(b) the equation …D 2 ‡ 2D ‡ 3†y ˆ 0 with the boundary conditions y ˆ 2,
Dy ˆ 0 when t ˆ 0;
(c) the equation …D 2 2D ‡ 1†y ˆ 0 with the boundary conditions y ˆ 5,
Dy ˆ 3 when t ˆ 0.
2.8 Find the particular integral of …D 2 ‡ 2D 1†y ˆ 3 ‡ t 3 .
2.9 Find the particular integral of …2D 2 ‡ 5D ‡ 7† ˆ3e 2t .
2.10 Find the particular integral of …3D 2 ‡ D 5†y ˆ cos 3t:
2.11 Simple harmonic motion of a pendulum (Fig. 2.4): Suspend a ball of mass m
at the end of a massless rod of length L and set it in motion swinging back
and forth in a vertical plane. Show that the equation of motion of the ball is
d 2
dt 2 ‡ g sin ˆ 0;
L
where g is the local gravitational acceleration. Solve this pendulum equation
for small displacements by replacing sin by .
2.12 Forced oscillations with damping: If we allow an external driving force F…t†
in addition to damping (Example 2.12), the motion of the oscillator is
governed by
y 00 ‡ b m y 0 ‡ k m y ˆ F…t†;
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