6.2. Solving the Homogeneous Part

Try these first

1. 1.

   What is the general form of the auxiliary equation for a second-order linear homogeneous differential equation?

   Hint

   Remember, it relates to the coefficients of the differential equation.

2. 2.

   What is the form of the complementary function if there are distinct real roots?

   Hint

   Think about how you combine the solutions of each root.

3. 3.

   What form does the complementary function take for distinct real roots?

   • y_h = (C_1 + C_2x)e^{r x}
   • y_h = C_1 e^{r_1 x} + C_2 e^{r_2 x}
   • y_h = e^{αx}(C_1 cos(βx) + C_2 sin(βx))

   Hint

   Focus on how these solutions are articulated.

4. 4.

   True or False: The form of the complementary function when roots are complex involves trigonometric functions.

   • True
   • False

   Hint

   Recall the connection of sin and cos with imaginary numbers.

5. 5.

   Solve the differential equation d²y/dx² - 5dy/dx + 6y = 0 and find the complete complementary function.

   Hint

   Start by setting the roots using the quadratic formula.

6. 6.

   For the equation d²y/dx² + 4y = 0, find the roots and formulate the complementary function.

   Hint

   Determine the imaginary components of the roots carefully.