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Chapter 4: Q4.3-20E (page 172)

Find a general solution.y'''-y''+2y=0

Short Answer

Expert verified

The general solution of the given equation y'''-y''+2y=0isy(t)=C1e-t+C2etcos(t)+C3etsin(t).

Step by step solution

01

Using rational root theorem.

First, you need to find the auxiliary equation and solve it. One has r3-r2+2=0.

The first divisor of role="math" localid="1654074445353" 2is 1 if 1will be one solution of the equation and (r-1) will be a factor.

That doesn't happen, but next, you can try with -1 so thatr+1 would be a factor.

02

Finding factor

Now you can divide r3-r2+2by r+1to get r2-2r+2.

Therefore, the equation can be factored as (r+1)(r2-2r+2)=0

Since r2-2r+2=0

role="math" localid="1654074733241" r=2±22-4×1×22r=1±i

03

Finding roots.

The roots of the auxiliary equation are r=-1,r=1+iand r=1-i

Thus, the general solution of the differential equation is:

y(t)=C1e-t+C2etcos(t)+C3etsin(t)

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Most popular questions from this chapter

RLCSeries Circuit. In the study of an electrical circuit consisting of a resistor, capacitor, inductor, and an electromotive force (see Figure), we are led to an initial value problem of the form

(20)LdIdt+RI+qC=E(t);q(0)=q0I(0)=I0,

whereL is the inductance in henrys,R is the resistance in ohms,C is the capacitance in farads, E(t)is the electromotive force in volts,q(t) is the charge in coulombs on the capacitor at the time t, androle="math" localid="1654852406088" I=dq/dt is the current in amperes. Find the current at time t if the charge on the capacitor is initially zero, the initial current is zero,role="math" localid="1654852401965" L=10H,R=20Ω,C=(6260)-1F , androle="math" localid="1654852397693" E(t)=100V .

Vibrating Spring with Damping. Using the model for a vibrating spring with damping discussed in Example3

(a)Find the equation of motion for the vibrating spring with damping ifm=10kg,b=60kg/sec,k=250kg/sec2,y(0)=0.3m,andy'(0)=-0.1m/sec.

(b)After how many seconds will the mass in part(a) first cross the equilibrium point?

(c)Find the frequency of oscillation for the spring system of part (a).

(d)Compare the results of problems32 and33determine what effect the damping has on the frequency of oscillation. What other effects does it have on the solution?

Find the solution to the initial value problem.

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