Chapter 11: Q.48 (page 860)

Find and graph the vector function determined $r (t) = ⟨ x (t), y (t) ⟩$ by the differential equation
$x^{'} (t) = - y, y^{'} (t) = x, x (0) = 1, y (0) = 0$ . ( HINT: What familiar pair of functions have the given properties ?)

Short Answer

Expert verified

Ans: The vector function is $r (t) = < \cos t, - \sin t >$

and the graph is,

Step by step solution

Step 1. Given information.

given

$x^{'} (t) = - y, y^{'} (t) = x, x (0) = 1, y (0) = 0$

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

$\begin{aligned} x^{'} (t) = - y \dots \dots (1 \\ y^{'} (t) = x \dots \dots (2) \\ x (0) = 1, y (0) = 0 \end{aligned}$

Differentiate equation (1) with respect to $t$

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

Substitute (2) in (3)

$\begin{aligned} x^{''} (t) = - x \\ x^{''} (t) + x = 0 \\ D^{2} x + x = 0 \\ (D^{2} + 1) x = 0 \end{aligned}$

Auxilary equation is

$m^{2} + 1 = 0 m = \pm i$

Therefore,

$x (t) = c_{1} \cos t + c_{2} \sin t$

At $t = 0, x (0) = 1$

$\begin{aligned} x (0) = c_{1} \cos (0) + c_{2} \sin (0) \\ 1 = c_{1} \end{aligned}$

Then

$x (t) = \cos t + c_{2} \sin t$

Differentiate with respect to $t$

role="math" localid="1649589790829" $\begin{aligned} x^{'} (t) = - \sin t + c_{2} \cos t \\ y (t) = - \sin t + c_{2} \cos t \end{aligned}$

Substitute $y (0) = 0$

role="math" localid="1649589878483" $0 = - \sin (0) + c_{2} \cos (0) c_{2} = 0$

Therefore,

x (t) = \cos t, y (t) = - \sin t

Thus, the vector function is

role="math" localid="1649590028804" $r (t) = < \cos t, - \sin t >$

Step 3. Draw the graph,

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Find and graph the vector function determined $r (t) = ⟨ x (t), y (t) ⟩$ by the differential equation
$x^{'} (t) = - y, y^{'} (t) = x, x (0) = 1, y (0) = 0$ . ( HINT: What familiar pair of functions have the given properties ?)

Short Answer

Step by step solution

Step 1. Given information.

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

Step 3. Draw the graph,

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Find and graph the vector function determined r(t)=⟨x(t),y(t)⟩by the differential equation x′(t)=−y,y′(t)=x,x(0)=1,y(0)=0. ( HINT: What familiar pair of functions have the given properties ?)

Short Answer

Step by step solution

Step 1. Given information.

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

Step 3. Draw the graph,

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Logic and Functions

Pure Maths

Probability and Statistics

Mechanics Maths

Geometry

Study anywhere. Anytime. Across all devices.

Find and graph the vector function determined $r (t) = ⟨ x (t), y (t) ⟩$ by the differential equation
$x^{'} (t) = - y, y^{'} (t) = x, x (0) = 1, y (0) = 0$ . ( HINT: What familiar pair of functions have the given properties ?)