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### Second-Order Linear Homogeneous Equations

#### Additional Examples

Solve the initial value problem
$$
\begin{align}
\frac{d^2y}{dx^2} - 10\frac{dy}{dx} + 74y &= 0 \\
y(0) &= 4 \\
y'(0) &= 5
\end{align}
$$

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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.*
$$
(D^2 - 10D + 74)y = 0
$$
*Step 2: Find the roots.*
We use the quadratic formula to compute the roots,
$ 5\pm 7i $

*Step 3: Write the general solution.*

$$ y(x) = c_1\exp(5x)\cos(7x) + c_2\exp(5x)\sin(7x) $$ 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) = 5c_1\exp(5x)\cos(7x) - 7c_1\exp(5x)\sin(7x) + 5c_2\exp(5x)\sin(7x) + 7c_2\exp(5x)\cos(7x).$ Then we plug in $ x=0 $ to get the following equations. $$ \begin{align} y(0) = c_1 &= 4 \\ y'(0) = 5c_1 + 7c_2 &= 5 \end{align} $$ We solve these equations to get $ c_1= 4 $ and $ c_2 = -15/7.$ Finally, we plug our values for $ c_1$ and $ c_2$ into the general solution to find our solution to the initial value problem.

$$
y(x) = 4\exp(5x)\cos(7x) - (15/7)\exp(5x)\sin(7x)
$$
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©2010, 2014 Andrew G. Bennett