1.3.1. General Form

Try these first

1. 1.

   Define a first-order linear differential equation.

   Hint

   Focus on the terms that make it first-order and linear.

2. 2.

   What is a non-homogeneous term?

   Hint

   Consider what additional factors affect the system.

3. 3.

   What form does a first-order linear differential equation take?

   • \\( \\frac{dy}{dx} + P(y)x = Q(x) \\)
   • \\( \\frac{dy}{dx} + P(x)y = Q(x) \\)
   • \\( dy + P(x)y = Q(y) \\)

   Hint

   Remember the relationship between the dependent variable and its rate of change.

4. 4.

   True or False: $Q(x)$ represents the homogeneous part of the equation.

   • True
   • False

   Hint

   Think about what distinguishes non-homogeneous from homogeneous.

5. 5.

   Given a real-world scenario, model it using a first-order linear differential equation. Then, outline the steps you would take to solve it using the integrating factor.

   Hint

   Identify the rate of change and the influences acting on your system.

6. 6.

   Create a physical scenario where the rate of a process varies due to external conditions, and derive the associated differential equation.

   Hint

   Consider how environmental factors might alter expected outcomes.