6.7. Non-Homogeneous Systems of Differential Equations

Try these first

1. 1.

   Define a non-homogeneous differential equation.

   Hint

   Think about what differentiates it from a homogeneous system.

2. 2.

   What are eigenvalues?

   Hint

   Remember, they are linked to the response characteristics of the system.

3. 3.

   What distinguishes a non-homogeneous system from a homogeneous system?

   • It has no external forces
   • It includes external terms
   • It is always stable

   Hint

   Think about what forces could be acting on a system.

4. 4.

   True or False: Eigenvalues can indicate the stability of a system.

   • True
   • False

   Hint

   Reflect on what eigenvalues represent in the context of systems.

5. 5.

   Analyze the following system and determine the eigenvalues: $\frac{dx}{dt} = x + 2y; \frac{dy}{dt} = -3x + 4y$. Find the solution to the homogeneous part.

   Hint

   Look for the determinant being equal to zero.

6. 6.

   Given the non-homogeneous equation $\frac{dx}{dt} = 2x + 3y + \cos(t); \frac{dy}{dt} = -xy + 4y + e^{-t}$, identify the external terms and solve the system.

   Hint

   Identify the non-homogeneous terms first!