Questions: Just as there are simultaneous algebraic equations (where a pair of numbers have to satisfy a pair of equations) there are systems of differential equations, (where a pair of functions have to satisfy a pair of differential equations). Indicate which pairs of functions satisfy this system. It will take some time to make all of the calculations. y₁' = 5/2 y₁ - 3/2 y₂ y₂' = -3/2 y₁ + 5/2 y₂ A. y₁=e^(-x) y₂=e^(-x) B. y₁=2 e^(-2 x) y₂=3 e^(-2 x) C. y₁=e^(x) y₂=e^(x) D. y₁=sin(x) y₂=cos(x) E. y₁=sin(x)+cos(x) y₂=cos(x)-sin(x) E. y₁=cos(x) y₂=-sin(x) G. y₁=e^(4 x) y₂=-e^(4 x) As you can see, finding all of the solutions, particularly of a system of equations, can be complicated and time consuming. It helps greatly if we study the structure of the family of solutions to the equations. Then if we find a few solutions will be able to predict the rest of the solutions using the structure of the family of solutions.

Solution Steps

To determine which pairs of functions satisfy the given system of differential equations, we need to substitute each pair into the equations and check if both equations are satisfied. For each pair, compute the derivatives \( y_1' \) and \( y_2' \), substitute them into the system, and verify if the left-hand side equals the right-hand side for both equations.

We are given a system of differential equations: \[ y_1' = \frac{5}{2} y_1 - \frac{3}{2} y_2 \] \[ y_2' = -\frac{3}{2} y_1 + \frac{5}{2} y_2 \]

We need to check which pairs of functions satisfy the system. The pairs are:

• A. \( y_1 = e^{-x}, \, y_2 = e^{-x} \)
• B. \( y_1 = 2e^{-2x}, \, y_2 = 3e^{-2x} \)
• C. \( y_1 = e^{x}, \, y_2 = e^{x} \)
• D. \( y_1 = \sin(x), \, y_2 = \cos(x) \)
• E. \( y_1 = \sin(x) + \cos(x), \, y_2 = \cos(x) - \sin(x) \)
• F. \( y_1 = \cos(x), \, y_2 = -\sin(x) \)
• G. \( y_1 = e^{4x}, \, y_2 = -e^{4x} \)

For each pair, substitute \( y_1 \) and \( y_2 \) into the differential equations and check if both equations are satisfied.

• Pair C: \( y_1 = e^{x}, \, y_2 = e^{x} \)

  • \( y_1' = e^{x}, \, y_2' = e^{x} \)
  • Substituting into the first equation: \( e^{x} = \frac{5}{2}e^{x} - \frac{3}{2}e^{x} \Rightarrow e^{x} = e^{x} \)
  • Substituting into the second equation: \( e^{x} = -\frac{3}{2}e^{x} + \frac{5}{2}e^{x} \Rightarrow e^{x} = e^{x} \)
  • Both equations are satisfied.

• Pair G: \( y_1 = e^{4x}, \, y_2 = -e^{4x} \)

  • \( y_1' = 4e^{4x}, \, y_2' = -4e^{4x} \)
  • Substituting into the first equation: \( 4e^{4x} = \frac{5}{2}e^{4x} + \frac{3}{2}e^{4x} \Rightarrow 4e^{4x} = 4e^{4x} \)
  • Substituting into the second equation: \( -4e^{4x} = -\frac{3}{2}e^{4x} - \frac{5}{2}e^{4x} \Rightarrow -4e^{4x} = -4e^{4x} \)
  • Both equations are satisfied.

Final Answer