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Chapter 4: Q3E (page 156)

Question: Show that if Fext(t)=0,m=1,k=9, and , then the equation has the "critically damped" solutionsy1(t)=e-3t andy2(t)=te-3t . What is the limit of these solutions as t鈫�鈭�?

Short Answer

Expert verified

Answer

They(t)=c1e-3t+c2te-3t solution is and the limit of the solution is .

Step by step solution

01

Finding the differential equation and the general solution.

The differential equation for the mass-spring oscillator is

my''+by'+ky=Fext.

GivenFext=0,m=1,k=9 and, then the differential equation isy''+6y'+9y=0.

The auxiliary equation for the given differential equation is,m2+6m+9=0(m+3)2=0

02

Check for critically damped.

If , then the system is critically damped. Here and , then;

b2-4ac=62-9脳4=0

So, the system is critically damped and, then the solutions are;

y(t)=c1e-3t+c2te-3t

If t鈫�鈭�, then e-3t=0. Therefore,

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