1.5.3. Example

Try these first

1. 1.

   Write the auxiliary equation for $y'' - 3y' + 2y = 0$.

   Hint

   Replace each derivative with powers of m.

2. 2.

   What is the general solution for $y'' - 4y = 0$?

   Hint

   Consider the auxiliary equation and the roots.

3. 3.

   What is the general form of the auxiliary equation derived from $y'' - 3y' + 2y = 0$?

   • $m^{2} - 3m + 2 = 0$
   • $m^{2} + 3m + 2 = 0$
   • $2m^{2} - 3m + 2 = 0$

   Hint

   Replace each derivative with m raised to its order.

4. 4.

   True or False: The general solution of a second-order linear homogeneous equation is a combination of two independent solutions.

   • True
   • False

   Hint

   Consider how solutions are structured in linear differential equations.

5. 5.

   Consider the equation $y'' + 4y = 0$. Find the roots of the auxiliary equation and determine the general solution. Discuss the implications of the roots.

   Hint

   Recall how to interpret complex roots in the general solution.

6. 6.

   Solve the differential equation $y'' + 12y' + 36y = 0$ and find particular values for $C_1$ and $C_2$ given initial conditions $y(0) = 2$ and $y'(0) = 0$.

   Hint

   Remember to differentiate the general solution when a double root is present to apply the initial conditions.