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Linear Algebra With Applications
Found in: Page 425
Linear Algebra With Applications

Linear Algebra With Applications

Book edition 5th
Author(s) Otto Bretscher
Pages 442 pages
ISBN 9780321796974

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Short Answer

Use the concept of a continuous dynamical systemdxdt=kx .Solve the differential equation dxdt=Ax. Solvethe system when A is diagonalizable over R,and sketch the phase portrait for 2 × 2 matrices A.

Solve the initial value problems posed in Exercises 1through 5. Graph the solution.

3. dPdt=0.03P with P(0)=7.

The solution is y=7e0.03t .

See the step by step solution

Step by Step Solution

Step 1: Definition of the differential equation

Consider the differential equation dydx=kx with initial value x0(k is an arbitrary constant). The solution is . x(t)=x0ekt

The solution of the linear differential equation dydx=kx and y(0)=C is .y=Cekx

Step 2: Calculation of the Solution

Given the differential equation dPdt=0.03P with the initial condition P(0)=7 .

Substitute in the solution y=Cekx as follows:


Hence, the solution for the differential equation dPdt=0.03P is y=7e0.03t.

Step 3: Graphical representation of the solution

The graph of the equation y=e0.03t is sketched below as follows:

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