Question

For the system \(x' = y; \, y' = -x\), the solution curve has the form:

• A. \(\frac{x^2}{c^2} - \frac{y^2}{c^2} = 1\)
• B. \(x^2 + y^2 = \frac{1}{2} c^2\)
• C. \(x^2 + y^2 = c^2\)
• D. \(-x^2 + y^2 = c^2\)

Hint:

Exam Tip:
For systems of ODEs:

• Eliminate the parameter by dividing the equations.
• Integrate to find the relationship between the variables.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
We have a system of first-order ODEs: \(x' = y\), \(y' = -x\).
We need to find the relationship between \(x\) and \(y\) (the solution curve).

Step 2: Key Formula or Approach:
Divide the two equations to eliminate the parameter \(t\): \[ \frac{dy}{dx} = \frac{dy/dt}{dx/dt} = \frac{-x}{y} \] So, \(y \, dy = -x \, dx\).

Step 3: Detailed Explanation:
Integrate: \[ \int y \, dy = -\int x \, dx \Rightarrow \frac{y^2}{2} = -\frac{x^2}{2} + c \] \[ x^2 + y^2 = 2c \] Let \(2c = c^2\) (absorbing the constant).
So, \(x^2 + y^2 = c^2\).
This is a circle centered at the origin.
Thus, the solution curve is \(x^2 + y^2 = c^2\).

Step 4: Final Answer:
Therefore, option (C) is correct.