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Chapter 4: Q4.3-12E (page 172)

Find a general solution u''+7u=0

Short Answer

Expert verified

The general solution of the given equation u''+7u=0is:

y(t)=(c1cos(7t)+c2sin(7t)).

Step by step solution

01

Differentiate the value of u.

Given differential equation isu''+7u=0

02

Finding roots of the auxiliary equation.

The auxiliary equation is r2+7e=0.

role="math" localid="1654069093148" r2+7=0r2=-7r=卤7i

03

Final answer.

Therefore, the general solution is:

y(t)=e0脳t(c1cos(7t)+c2sin(7t))y(t)=(c1cos(7t)+c2sin(7t))

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Most popular questions from this chapter

Question: Find a synchronous solution of the form 础肠辞蝉惟迟+叠蝉颈苍惟迟to the given forced oscillator equation using the method of Example to solve forAand By''+2y'+5y=-50sin5t,惟=5 .

.

Find a particular solution to the differential equation.

y''+2y'+2y=4te-tcost

A 14鈥� kg mass is attached to a spring with stiffness 8 N/m. The damping constant for the system is 14N-sec/m. If the mass is moved 1 m to the left of equilibrium and released,what is the maximum displacement to the right that it will attain?

Speed Bumps. Often bumps like the one depicted in Figure 4.11 are built into roads to discourage speeding. The figure suggests that a crude model of the vertical motion y(t) of a car encountering the speed bump with the speed V is given by

y(t)=0 for t鈮�-L2V

localid="1655121580511" my''+ky={F0cos(蟺痴迟L),鈥夆赌�for鈥�|t|<L2V0,鈥夆赌夆�夆赌夆��for鈥�t鈮�L2V}

(The absence of a damping term indicates that the car鈥檚 shock absorbers are not functioning.)

  1. Taking m=k=1,鈥夆赌�L=蟺, andF0=1 in appropriate units, solve this initial value problem. Thereby showing that the formula for the oscillatory motion after the car has traversed the speed bump is y(t)=Asint, where the constant A depends on the speed V.
  2. Plot the amplitude |A| of the solution y(t) found in part (a) versus the car鈥檚 speed V. From the graph, estimate the speed that produces the most violent shaking of the vehicle.

An 8-kg mass is attached to a spring hanging from the ceiling and allowed to come to rest. Assume that the spring constant is 40 N/m and the damping constant is 3 N/sec. At time t = 0, an external force 2sin2t+蟺4N is applied to the system. Determine the amplitude and frequency of the steady-state solution.
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