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### Second-Order Linear Homogeneous Equations

#### Additional Examples

Solve the initial value problem
$$
\begin{align}
\frac{d^2y}{dx^2} - 3\frac{dy}{dx} - 28y &= 0 \\
y(0) &= -10 \\
y'(0) &= 3
\end{align}
$$

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©2010, 2014 Andrew G. Bennett

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This is a second-order linear constant-coefficient initial value problem. First we find the general solution.

*Step 1: Write the equation in operator form.*
$$
(D^2 - 3D - 28)y = 0
$$
*Step 2: Find the roots.*
The equation factors into $(D + 4)(D - 7)y = 0.$
So our roots are -4 and 7.

*Step 3: Write the general solution.*

$$ y(x) = c_1\exp(-4x) + c_2\exp(7x) $$ Now that we have the general solution, we plug in our initial values to find the solution to the initial value problem. First we compute, $ y'(x) = -4c_1\exp(-4x) + 7c_2\exp(7x).$ Then we plug in $ x=0 $ to get the following equations. $$ \begin{align} y(0) = c_1 + c_2 &= -10 \\ y'(0) = -4c_1 + 7c_2 &= 3 \end{align} $$ We solve these equations to get $ c_1= -73/11 $ and $ c_2 = -37/11.$ Finally, we plug our values for $ c_1$ and $ c_2$ into the general solution to find our solution to the initial value problem.

$$
y(x) = -(73/11)\exp(-4x) - (37/11)\exp(7x)
$$
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©2010, 2014 Andrew G. Bennett