Problem 5 - Calculus for Biology and Medicine

Question

Consider $$ \begin{array}{l} \frac{d x_{1}}{d t}=-x_{1}+2 x_{2} \\ \frac{d x_{2}}{d t}=x_{1} \end{array} $$ Determine the direction vectors associated with the following points in the \(x_{1}-x_{2}\) plane, and graph the direction vectors in the \(x_{1}-\) \(x_{2}\) plane: \((1,0),(0,1),(-1,0),(0,-1),(1,1),(0,0)\), and \((-2,1)\)

Step-by-Step Solution

Answer

Direction vectors are: \((-1,1),(2,0),(1,-1),(-2,0),(1,1),(0,0),(4,-2)\). Plot them with tails at given points.

Step 1: Understanding the System of Differential Equations

The given system of differential equations is \( \frac{d x_{1}}{d t} = -x_{1} + 2x_{2} \) and \( \frac{d x_{2}}{d t} = x_{1} \). These describe how \( x_{1} \) and \( x_{2} \) change over time. We need to determine the direction vectors \((\frac{d x_{1}}{d t}, \frac{d x_{2}}{d t})\) at specific points in the \( x_{1}-x_{2} \) plane.

Step 2: Calculate Direction Vectors for Each Point

For every point \((x_{1}, x_{2})\), substitute the values into the differential equations to find the direction vector:- **At (1, 0):** \( \frac{d x_{1}}{d t} = -1 + 0 = -1 \), \( \frac{d x_{2}}{d t} = 1 \). Direction vector: \((-1, 1)\).- **At (0, 1):** \( \frac{d x_{1}}{d t} = 0 + 2 = 2 \), \( \frac{d x_{2}}{d t} = 0 \). Direction vector: \((2, 0)\).- **At (-1, 0):** \( \frac{d x_{1}}{d t} = 1 + 0 = 1 \), \( \frac{d x_{2}}{d t} = -1 \). Direction vector: \((1, -1)\).- **At (0, -1):** \( \frac{d x_{1}}{d t} = 0 - 2 = -2 \), \( \frac{d x_{2}}{d t} = 0 \). Direction vector: \((-2, 0)\).- **At (1, 1):** \( \frac{d x_{1}}{d t} = -1 + 2 = 1 \), \( \frac{d x_{2}}{d t} = 1 \). Direction vector: \((1, 1)\).- **At (0, 0):** Both derivatives are zero, so the direction vector is \((0, 0)\).- **At (-2, 1):** \( \frac{d x_{1}}{d t} = 2 + 2 = 4 \), \( \frac{d x_{2}}{d t} = -2 \). Direction vector: \((4, -2)\).

Step 3: Graphing the Direction Vectors

Plot each point in the \( x_{1}-x_{2} \) plane and draw the corresponding direction vector by considering the point as the tail of the vector. For example, at point \((1, 0)\), draw a vector with tail at (1, 0) and head at \((1, 0) + (-1, 1) = (0, 1)\). Repeat this process for all points with their respective vectors.

Key Concepts

Direction Vectors

A direction vector helps us visualize how a point in the \(x_1-x_2\) plane changes over time according to given differential equations. These vectors are essentially arrows that point in the direction of change. Think of them as guiding arrows that tell us which way a point wants to "move."

• The direction vector at each point is obtained by substituting the point's coordinates \(x_1, x_2\) into the system of differential equations provided.
• For example, if you substitute a point like \(x_1 = 1\) and \(x_2 = 0\) into the equations \(\frac{d x_1}{d t}=-x_1+2x_2\) and \(\frac{d x_2}{d t}=x_1\), you find that \(\frac{d x_1}{d t}=-1\) and \(\frac{d x_2}{d t}=1\). Hence, the direction vector at this point is \(-1, 1\).

Understanding direction vectors is key to interpreting how systems change dynamically over time and help in effectively sketching direction fields in the plane.

System of Equations

A system of differential equations consists of multiple equations that describe how variables change with respect to one another over time. In our example, \(\frac{d x_1}{d t}=-x_1+2x_2\) and \(\frac{d x_2}{d t}=x_1\) form such a system.

• This system is interconnected, meaning the change in one variable, \(x_1\), affects the other, \(x_2\), and vice versa. In these equations, the rate of change of \(x_1\) depends on both \(x_1\) and \(x_2\), while the rate of change of \(x_2\) depends solely on \(x_1\).
• Solving such systems usually involves finding solutions that identify how the variables evolve together through time. This highlights interdependencies between variables.

This relationship makes systems of differential equations powerful tools for modeling real-world phenomena such as population dynamics, chemical reactions, and electrical circuits.

Graphing in the Plane

Graphing direction vectors in a plane involves plotting points and drawing vectors that represent their direction of change. This visualization provides insight into the behavior of the system over time.

• To draw a direction vector, first plot the given point \(x_1, x_2\) on the plane. The vector starts at this point, which serves as its tail.
• The vector itself \(dx_1/dt, dx_2/dt\) is an arrow pointing from the tail to the head determined by summing the point coordinates with the direction vector.

For example, at \( (1, 0) \), you plot the point and draw a vector from \( (1, 0) \) to \( (0, 1) \). This process is repeated for other points to create a field of arrows, showing how the system behaves overall. Graphing these helps in visualizing the flow and stability of solutions in the dynamic system.