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Linear Algebra With Applications

Otto Bretscher

$Math Studyset 91影视 Explanations$ Math

5th Edition 路 442 pages

ISBN: 9780321796974

Chapter 9: Q1E (page 425)

Use the concept of a continuous dynamical system.Solve the differential equation $\frac{d x}{d t} = 鈭� k x$ . Solvethe system $\frac{d \overset{鈫�}{x}}{d t} = A \overset{鈫�}{x}$ whenAis diagonalizable overR,and sketch the phase portrait for 2脳2 matricesA.
Solve the initial value problems posed in Exercises 1through 5. Graph the solution.
$\frac{d x}{d t} = 5 x$ with $x (0) = 7$ .

Short Answer

Expert verified

The solution is $y = 7 e^{5 t}$ .

Step by step solution

01

Definition of the differential equation

Consider the differential equation $\frac{d y}{d x} = k x$ with initial value $x_{0}$ (k is an arbitrary constant). The solution isrole="math" localid="1659551917942" $x (t) = x_{0} e^{k t}$ .

The solution of the linear differential equation $\frac{d y}{d x} = k x$ and $y (0) = C$ is $y = {Ce}^{kx}$ .

02

Calculation of the solution

Given the differential equation $\frac{d x}{d t} = 5 x$ with the initial condition $x (0) = 7$ .

Substitute in the solution $y = C e^{k x}$ as follows.

$\begin{matrix} y = C e^{k x} \\ y = 7 e^{5 t} \end{matrix}$

Hence, the solution for the differential equation $\frac{d x}{d t} = 5 x$ is $y = 7 e^{5 t}$ .

03

Graphical representation the equation

The graph of the equation $y = 7 e^{5 t}$ is sketched below as follows:

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Most popular questions from this chapter

Find the real solution of the system $\frac{d \overset{鈫�}{x}}{d t} = [\begin{matrix} 2 & 4 \\ - 4 & 2 \end{matrix}] \overset{鈫�}{x}$

Solve the nonlinear differential equations in Exercises 6through 11 using the method of separation of variables:Write the differential equation $\frac{d x}{d t} = f x$ as $\frac{d x}{f x} = d t$ and integrate both sides.

10. $\frac{d x}{d t} = \frac{1}{c o s (x)}, x (0) = 0$

Consider the system $\frac{d \overset{鈫�}{x}}{d t} = (\begin{matrix} 0 & 1 \\ a & b \end{matrix}) \overset{鈫�}{x}$ where a and b are arbitrary constants for which values of a and b is the zero state a stable equilibrium solution?

Find the real solution of the system $\frac{d \overset{鈫�}{x}}{d t} = [\begin{matrix} 0 & - 3 \\ 3 & 0 \end{matrix}] \overset{鈫�}{x}$

For the linear system $\frac{d \overset{鈫�}{x}}{d t} = [\begin{matrix} 3 & 0 \\ - 2.5 & 0.5 \end{matrix}] \overset{鈫�}{x} (t)$ find the matching phase portrait.

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