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Chapter 11: Q.48 (page 860)

Find and graph the vector function determined r(t)=鉄�x(t),y(t)鉄�by the differential equation

x鈥�(t)=鈭�y,y鈥�(t)=x,x(0)=1,y(0)=0. ( HINT: What familiar pair of functions have the given properties ?)

Short Answer

Expert verified

Ans: The vector function is r(t)=<cost,-sint>

and the graph is,

Step by step solution

01

Step 1. Given information.

given

x鈥�(t)=鈭�y,y鈥�(t)=x,x(0)=1,y(0)=0

02

Step 2. The objective of this problem is to find and graph the vector r(t)=⟨x(t),y(t)⟩ determined by the differential equations

x鈥�(t)=鈭�y鈥�鈥�(1y鈥�(t)=x鈥�鈥�(2)x(0)=1,y(0)=0

Differentiate equation (1) with respect to t

x''(t)=-y'(t)....(3)

Substitute (2) in (3)

x鈥测赌�(t)=鈭�xx鈥测赌�(t)+x=0D2x+x=0D2+1x=0

Auxilary equation is

m2+1=0m=卤i

Therefore,

x(t)=c1cos鈦�t+c2sin鈦�t

At t=0,x(0)=1

x(0)=c1cos鈦�(0)+c2sin鈦�(0)1=c1

Then

x(t)=cost+c2sint

Differentiate with respect to t

role="math" localid="1649589790829" x鈥�(t)=鈭�sin鈦�t+c2cos鈦�ty(t)=鈭�sin鈦�t+c2cos鈦�t

Substitutey(0)=0

role="math" localid="1649589878483" 0=鈭�sin鈦�(0)+c2cos鈦�(0)c2=0

Therefore,

x(t)=cost,y(t)=-sint

Thus, the vector function is

role="math" localid="1649590028804" r(t)=<cost,-sint>

03

Step 3. Draw the graph,

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Most popular questions from this chapter

In Exercises 19鈥�21 sketch the graph of a vector-valued function r(t)=x(t),y(t) with the specified properties. Be sure to indicate the direction of increasing values oft.

Domainlocalid="1649578696830" t鈮�0,r0=0,3,r1=-2,1,limt鈫�鈭�x(t)=-5,andlimt鈫�鈭�y(t)=-鈭�.

Under what conditions does a twice-differentiable vector valued function r(t) not have a binormal vector at a point in the domain of r(t)?

For each of the vector-valued functions, find the unit tangent vector.

r(t)=(cos伪t,sin伪t)

Under what conditions does a differentiable vector-valued functionr(t) not have a unit tangent vector at a point in the domain of r(t)?

For each of the vector-valued functions in Exercises 22鈥�28, find the unit tangent vector.

r(t)=(cos3t,sin3t)
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