MA350
Di¤erential Equations I
Revision III: Chapters 3.8, 4.1-4.3 and 7.1-7.3
1. In Section 3.8, you should know how to solve the problem mechanial vibration if a mass is attached to
a spring. This system is described by the following di¤erential equations
mu00 +u0+ku = 0; u (0) = u0and u0(0) = v0
where u(t)is the position of the box, mis the "mass" of the box, is the damping constant, kis
the spring’s constant, u0is the initial position, and v0is the initial velocity. In the undamped free
vibration, = 0. The spring keeps on vibrating. You should know how to determine the amplitude,
period, frequency, and phase of the motion. When 24km = 0, this case is called critical damping.
When 24km > 0, this case is identi…ed as overdamped. If 24km < 0, this case is named damped
vibration. In a LRC circuit, if there is an initial charge Q0stored in the capacitor and the current
along the circuit is I0, the charge stores in the capacitor satis…es the equation
Ld2Q
dt2+RdQ
dt +Q
C= 0; Q (t0) = Q0and Q0(t0) = I0:
(a) i. A mass weighing 16 lb stretches a spring 3inch. If the mass is pushed downward, the spring
is extended a distance of 2inches, and then set in motion with a initial downward velocity of
3ft/sec. If the damping coe¢ cient is 4lb-sec/ft, …nd the position uof the mass at any time
twhen g= 32 lb-sec2/ft.
ii. Suppose that the damping coe¢ cient is 0, …nd the amplitude, period, frequency, and phase
of the motion of the mass.
(b) In a LRC circuit, if a series circuit has a capacitor 410farad, a resistor of R= 50 ohms, and
an inductor of 0:0002 henry. The initial charge on the capacitor is 3104coulomb and there is
an initial current 0:1ampere. Find the charge Qon the capacitor at any time t.
2. Find the Wronskian of the functions t2+t; etamd e2t. Then, determine whether these functions are
linear dependent or independent.
3. Know how to solve the general solution of homogeneous di¤erential equations, for example, solve
y000 5y00 32y036y= 0:
Then, …nd the solution of the following initial value problem
y000 5y00 32y036y= 0; y (0) = 8; y0(0) = 4; y00 (0) = 10:
4. Know how to …nd undetermined co e¢ cients to …nd the form of the particular solution of a nonhomo-
geneous di¤erential equations, for example, solve the particular solution of the following di¤erential
equations (i)
y000 5y00 32y036y= 25x+ 3e9xxe2x;
(ii)
y(4) + 10y00 + 9y=tsin 3t+ cos t:
5. Change an nth order di¤erential equation into a system of …rst order di¤erential equations and vice
versa. For example, the di¤erential equation y(n)=ft; y ; y0; :::; y(n1)can be transform to x1=y,
x2=y0, ..., xn=y(n1) and x0
n=f(t; x1; x2; :::; xn). For example, write the following di¤erential
equation into a system of …rst order di¤erential equations (i)
y00 + 5y0+ 7y=e8t;
(ii)
y(4) + 7y000 9t2y00 + 10y07y= sin t:
