1.3.3. Example

Try these first

1. 1.

   What is the form of a first-order linear differential equation?

   Hint

   Look for terms involving dy/dx, a function of y, and a separate function.

2. 2.

   Identify P(x) from the equation dy/dx + 2y = e^(-x).

   Hint

   It's the coefficient of y in the equation.

3. 3.

   What is the purpose of the integrating factor in differential equations?

   • To simplify the right side
   • To combine terms
   • To rewrite the equation as an exact derivative

   Hint

   Think about what converting the equation into an exact derivative means.

4. 4.

   True or False: The integrating factor is always e^(∫P(x) dx).

   • True
   • False

   Hint

   Recall how we derived the integrating factor.

5. 5.

   Solve the equation dy/dx + 4y = sin(x).

   Hint

   Consider the integration techniques for sin(x)e^(4x).

6. 6.

   Derive the general solution to the differential equation dy/dx + 5y = 3e^(3x).

   Hint

   Remember to simplify your final answer.