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Problem (2): Solving Homogeneous Second Order Differential Equations Find the solution to the following differential equations. a) $y'' - 6y' - 2y = 0$ b) $y'' + 16y = 0$ $y(frac{pi}{2}) = -10$ $y'(frac{pi}{2}) = 3$ c) $y'' + 14y' + 49y = 0$ $y(-4) = -1$ $y'(-4) = 5$ Hint: We have three cases that we need to look at and this will be addressed differently in each of these cases. So, what are the cases? As we previously noted the characteristic equation is quadratic and so will have two roots, $r_1$ and $r_2$. The roots will have three possible forms. These are 1. Real, distinct roots, $r_1 eq r_2$. 2. Complex root, $r_{1,2} = lambda pm mu i$. 3. Double roots, $r_1 = r_2 = r$.

          Problem (2): Solving Homogeneous Second Order Differential Equations
Find the solution to the following differential equations.
a) $y'' - 6y' - 2y = 0$
b) $y'' + 16y = 0$  $y(frac{pi}{2}) = -10$  $y'(frac{pi}{2}) = 3$
c) $y'' + 14y' + 49y = 0$  $y(-4) = -1$  $y'(-4) = 5$
Hint:
We have three cases that we need to look at and this will be addressed differently in each of these cases.
So, what are the cases? As we previously noted the characteristic equation is quadratic and so will have
two roots, $r_1$ and $r_2$. The roots will have three possible forms. These are
1. Real, distinct roots, $r_1 
eq r_2$.
2. Complex root, $r_{1,2} = lambda pm mu i$.
3. Double roots, $r_1 = r_2 = r$.

$Problem (2): Solving Homogeneous Second Order Differential Equations Find the solution to the following differential equations. a) y” - 6y' - 2y = 0 b) y” + 16y = 0 y(fracpi2) = -10 y'(fracpi2) = 3 c) y” + 14y' + 49y = 0 y(-4) = -1 y'(-4) = 5 Hint: We have three cases that we need to look at and this will be addressed differently in each of these cases. So, what are the cases? As we previously noted the characteristic equation is quadratic and so will have two roots, r1 and r2. The roots will have three possible forms. These are 1. Real, distinct roots, r1 eq r2. 2. Complex root, r1,2 = lambda pm mu i. 3. Double roots, r1 = r2 = r.$

Added by Thomas D.

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