To analyze the functions \( y_1 \) and \( y_2 \) defined by the differential equations \( \frac{dy_1}{dx} = -y_2 \) and \( \frac{dy_2}{dx} = y_1 \), along with the initial conditions \( y_1(0) = 1 \) and \( y_2(0) = 0 \), we can derive their relationship and determine the geometric shape represented by the set \( S = \{(y_1(x), y_2(x)) : x \in \mathbb{R}\} \).
Understanding the System of Equations
We start with the given system of first-order differential equations. The first equation tells us that the rate of change of \( y_1 \) is equal to the negative of \( y_2 \), while the second equation indicates that the rate of change of \( y_2 \) is equal to \( y_1 \). This suggests a relationship between the two functions that may lead to a circular or oscillatory behavior.
Finding the Solutions
To solve this system, we can differentiate the first equation with respect to \( x \):
- From \( \frac{dy_1}{dx} = -y_2 \), we differentiate to get \( \frac{d^2y_1}{dx^2} = -\frac{dy_2}{dx} \).
- Substituting \( \frac{dy_2}{dx} = y_1 \) into this gives us \( \frac{d^2y_1}{dx^2} = -y_1 \).
This is a second-order linear differential equation with constant coefficients, which has the characteristic equation \( r^2 + 1 = 0 \). The solutions to this equation are \( r = i \) and \( r = -i \), leading to the general solution:
y_1(x) = A \cos(x) + B \sin(x)
where \( A \) and \( B \) are constants determined by initial conditions. Now, using the first equation, we can find \( y_2 \):
y_2(x) = \frac{dy_1}{dx} = -A \sin(x) + B \cos(x)
Applying Initial Conditions
Next, we apply the initial conditions to find \( A \) and \( B \):
- From \( y_1(0) = 1 \): \( A \cos(0) + B \sin(0) = A = 1 \).
- From \( y_2(0) = 0 \): \( -A \sin(0) + B \cos(0) = B = 0 \).
Thus, we have:
y_1(x) = \cos(x)
y_2(x) = \sin(x)
Characterizing the Set S
Now that we have explicit forms for \( y_1 \) and \( y_2 \), we can analyze the set \( S \). The pair \( (y_1(x), y_2(x)) = (\cos(x), \sin(x)) \) describes a point on the unit circle in the Cartesian plane, since:
y_1^2 + y_2^2 = \cos^2(x) + \sin^2(x) = 1
This equation represents a circle with a radius of 1 centered at the origin.
Conclusion
In summary, the set \( S \) defined by the functions \( y_1 \) and \( y_2 \) lies on a circle. The relationship between \( y_1 \) and \( y_2 \) is a classic example of parametric equations describing circular motion, illustrating how differential equations can reveal geometric properties of solutions.
