3.3.2. Case 2: Repeated Real Roots (D =0)

Try these first

1. 1.

   Write the general solution for a differential equation with repeated roots.

   Hint

   Use the form that includes a polynomial term.

2. 2.

   What does a discriminant of zero indicate?

   Hint

   Think about the nature of roots in polynomials.

3. 3.

   What form does the general solution take for repeated roots?

   • y(x) = C_1 e^{rx}
   • y(x) = (C_1 + C_2 x)e^{rx}
   • y(x) = C_1 e^{r_1 x} + C_2 e^{r_2 x}

   Hint

   Think about the role of x in the solution.

4. 4.

   Is the discriminant zero when the roots are complex?

   • True
   • False

   Hint

   Recall the conditions for different types of roots.

5. 5.

   Given the differential equation y'' - 6y' + 9y = 0, analyze how varying initial conditions of y(0) = 5 and y'(0) = 0 affect the constants and the resultant function.

   Hint

   Apply the initial conditions directly for each constant.

6. 6.

   Discuss in detail how repeated roots could relate to real-world scenarios, for instance, in the context of earthquake-resistant structures.

   Hint

   Consider how structures respond to forces over time with repeated root solutions.