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Q3E

Expert-verifiedFound in: Page 425

Book edition
5th

Author(s)
Otto Bretscher

Pages
442 pages

ISBN
9780321796974

**Use the concept of a continuous dynamical system$\frac{\mathbf{d}\mathbf{x}}{\mathbf{d}\mathbf{t}}{\mathbf{=}}{\mathbf{-}}{\mathit{k}}{\mathit{x}}$ .Solve the differential equation $\frac{\mathbf{d}\overrightarrow{\mathbf{x}}}{\mathbf{d}\mathbf{t}}{\mathbf{=}}{\mathit{A}}\overrightarrow{\mathbf{x}}$. 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. $\frac{\mathbf{d}\mathbf{P}}{\mathbf{d}\mathbf{t}}{\mathbf{=}}{\mathbf{0}}{\mathbf{.}}{\mathbf{03}}{\mathit{P}}$ with ${\mathit{P}}\mathbf{\left(}\mathbf{0}\mathbf{\right)}{\mathbf{=}}{\mathbf{7}}$**.

The solution is $y=7{e}^{0.03t}$ **.**

**Consider the differential equation $\frac{dy}{dx}{=}{k}{x}$ with initial value ${{x}}_{{0}}$(k is an arbitrary constant). The solution is . ${x}\left(t\right){=}{{x}}_{{0}}{{e}}^{kt}$**

** **

**The solution of the linear differential equation $\frac{dy}{dx}{=}{k}{x}$**** and ${\text{y}}\left(0\right){=}{\text{C}}$ is .${\text{y}}{=}{{\text{Ce}}}^{{\text{kx}}}$**

Given the differential equation $\frac{dP}{dt}=0.03P$ with the initial condition $P\left(0\right)=7$ .

Substitute in the solution $y=C{e}^{kx}$ as follows:

$\begin{array}{l}y=C{e}^{kx}\\ y=7{e}^{0.03t}\end{array}$

Hence, the solution for the differential equation $\frac{dP}{dt}=0.03P$ is $y=7{e}^{0.03t}$.

The graph of the equation $y={e}^{0.03t}$ is sketched below as follows:

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