Q23E Consider a linear system dx鈫抎t... [FREE SOLUTION]

91影视

Linear Algebra With Applications

Otto Bretscher

$Math Studyset 91影视 Explanations$ Math

5th Edition 路 442 pages

ISBN: 9780321796974

Chapter 9: Q23E (page 426)

Consider a linear system $\frac{d \overset{鈫�}{x}}{dt} = A \overset{鈫�}{x}$ of arbitrary size. Suppose ${\overset{鈫�}{x}}_{1} (t)$ is a solution of the system and k is an arbitrary constant. Is $\overset{鈫�}{x} (t) = k {\overset{鈫�}{x}}_{1} (t)$ a solution as well? How do you know?

Short Answer

Expert verified

Yes, $\overset{鈫�}{x} (t) = k {\overset{鈫�}{x}}_{1} (t)$ would be the solution as well since $\frac{d (k {\overset{鈫�}{x}}_{1})}{d t} = k \frac{d {\overset{鈫�}{x}}_{1}}{d t}$ .

Step by step solution

The matrix form of the derivatives

The rates of change of the components of the state vector $\overset{鈫�}{x} (t)$ , or its derivatives $\frac{d x_{1}}{d t}, \frac{d x_{2}}{d t}, 鈥�, \frac{d x_{n}}{d t}$ .

When these rates depend linearly on $x_{1}, x_{2}, 鈥�, x_{n}$ , write the matrix equation as $\frac{d \overset{鈫�}{x}}{d t} = A \overset{鈫�}{x}$

Check whether x→t=kx→1tis a solution as well

When ${\overset{鈫�}{x}}_{1} (t)$ would be a solution of the linear system $\frac{d \overset{鈫�}{y}}{d t} = A \overset{鈫�}{y}$ would satisfy this matrix equation.

Consider k as an arbitrary constant. Utilize the linearity of the differentiation as follow:

$\begin{array}{rcl} \frac{d \overset{鈫�}{x}}{d t} & = & \frac{d (k {\overset{鈫�}{x}}_{1})}{d t} \\ = & k \frac{d {\overset{鈫�}{x}}_{1}}{d t} \\ = & k A {\overset{鈫�}{x}}_{1} \\ = & A k {\overset{鈫�}{x}}_{1} \\ = & A \overset{鈫�}{x} \end{array}$

Therefore, $\overset{鈫�}{x} (t) = k {\overset{鈫�}{x}}_{1} (t)$ would be a solution of the given system.

Thus, yes, $\overset{鈫�}{x} (t) = k {\overset{鈫�}{x}}_{1} (t)$ would be the solution as well since $\frac{d (k {\overset{鈫�}{x}}_{1})}{d t} = k \frac{d {\overset{鈫�}{x}}_{1}}{d t}$ .

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91影视

Short Answer

Step by step solution

The matrix form of the derivatives

Check whether x→t=kx→1tis a solution as well

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91影视

Consider a linear system dx鈫�dt=Ax鈫� of arbitrary size. Supposex鈫�1(t) is a solution of the system and k is an arbitrary constant. Is x鈫�t=kx鈫�1ta solution as well? How do you know?

Short Answer

Step by step solution

The matrix form of the derivatives

Check whether x→t=kx→1tis a solution as well

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Probability and Statistics

Geometry

Decision Maths

Theoretical and Mathematical Physics

Applied Mathematics

Mechanics Maths

Study anywhere. Anytime. Across all devices.