2.4. Case I: Real and Distinct Roots

Try these first

1. 1.

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

   Hint

   Look for terms involving y and its derivatives.

2. 2.

   Define a homogeneous differential equation.

   Hint

   Focus on the equality condition.

3. 3.

   What form does the general solution take when the roots are real and distinct?

   • y(x) = C1 e^(m1 x) + C2 e^(m2 x)
   • y(x) = (C1 + C2 x)e^(m x)
   • y(x) = e^(αx)(C cos(βx) + C sin(βx))

   Hint

   Recall the response of distinct roots.

4. 4.

   True or False: The solution to a second-order linear homogeneous equation can always be expressed using two arbitrary constants.

   • True
   • False

   Hint

   Think about how we determine specific solutions.

5. 5.

   Given the equation d²y/dx² + 4dy/dx + 5y = 0, determine if the roots are real and distinct. Solve for y(x).

   Hint

   Calculate the discriminant.

6. 6.

   If the auxiliary equation yields roots m1 = 3 and m2 = 5, find the specific solution if given initial conditions y(0) = 1, y'(0) = 0.

   Hint

   Express C1 in terms of C2 using the first initial condition.