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Chapter 9: Q3E (page 425)

Use the concept of a continuous dynamical systemdxdt=鈭�kx.Solve the differential equation dx鈫�dt=Ax鈫�. Solvethe system when Ais 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.

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

Short Answer

Expert verified

The solution isy=7e0.03t.

Step by step solution

01

Definition of the differential equation

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

The solution of the linear differential equationdydx=kxandy(0)=C is .y=Cekx

02

Calculation of the Solution

Given the differential equationdPdt=0.03P with the initial conditionP(0)=7 .

Substitute in the solutiony=Cekxas follows:

y=Cekxy=7e0.03t

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

03

Graphical representation of the solution

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

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