Chapter 11: Q. 46 (page 860)

Find and graph the vector function determined $r (t) = ⟨ x (t), y (t) ⟩$ by the differential equation
$x^{'} (t) = x, y^{'} (t) = x^{2}, x (0) = 1, y (0) = 2$ . (HINT: Start by solving the initial-value problem $x^{'} (t) = x, x (0) = 1$ .)

Short Answer

Expert verified

Ans: Vector function is $r (t) = < e^{t}, t e^{2 t} + 2 >$

Graph is:

Step by step solution

Step 1. Given information.

given,

$x^{'} (t) = x, y^{'} (t) = x^{2}, x (0) = 1, y (0) = 2$

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

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

solve equation $1$

$x' (t) = x$

$\frac{d x}{d t} = x$

Separate the variables

$\frac{d x}{x} = d t$

Integrate both sides

$\begin{aligned} \int \frac{d x}{x} = \int d t + c_{1} \\ \ln x = t + c_{1} \\ x = e^{t + c_{1}} \\ x = c e^{t} \end{aligned}$

Therefore,

$x = e^{t} . . . . . (3)$

Step 3. Now solve

$\begin{aligned} y^{'} (t) = x^{2} \\ \frac{d y}{d t} = x^{2} \\ d y = x^{2} d t \end{aligned}$

Integrate

$\begin{aligned} \int d y = \int x^{2} d t + c_{2} \\ y = x^{2} t + c_{2} \\ At t = 0, y = 2 \\ 2 = c_{2} \end{aligned}$

Therefore,

$y = x^{2} t + 2 . . . . (4)$

Use $x = e^{t}$ from (3) into (4)

$y = t e^{2 t} + 2$

Thus, the vector of the function is

$r (t) = ⟨ e^{t}, t e^{2 t} + 2 ⟩$

Step 4. Draw the graph

Graph is:

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Find and graph the vector function determined $r (t) = ⟨ x (t), y (t) ⟩$ by the differential equation
$x^{'} (t) = x, y^{'} (t) = x^{2}, x (0) = 1, y (0) = 2$ . (HINT: Start by solving the initial-value problem $x^{'} (t) = x, x (0) = 1$ .)

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. Now solve

Step 4. 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)=x,y′(t)=x2,x(0)=1,y(0)=2. (HINT: Start by solving the initial-value problem x′(t)=x,x(0)=1.)

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. Now solve

Step 4. Draw the graph

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Mechanics Maths

Applied Mathematics

Decision Maths

Probability and Statistics

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Geometry

Study anywhere. Anytime. Across all devices.

Find and graph the vector function determined $r (t) = ⟨ x (t), y (t) ⟩$ by the differential equation
$x^{'} (t) = x, y^{'} (t) = x^{2}, x (0) = 1, y (0) = 2$ . (HINT: Start by solving the initial-value problem $x^{'} (t) = x, x (0) = 1$ .)