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### Second-Order Linear Homogeneous Equations

#### Additional Examples

Solve the initial value problem
$$
\begin{align}
\frac{d^2y}{dx^2} - 7\frac{dy}{dx} + 6y &= 0 \\
y(0) &= -10 \\
y'(0) &= 3
\end{align}
$$

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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.*
$$
(D^2 - 7D + 6)y = 0
$$
*Step 2: Find the roots.*
The equation factors into $(D - 1)(D - 6)y = 0.$
So our roots are 1 and 6.

*Step 3: Write the general solution.*

$$ y(x) = c_1\exp(x) + c_2\exp(6x) $$ 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) = c_1\exp(x) + 6c_2\exp(6x).$ Then we plug in $ x=0 $ to get the following equations. $$ \begin{align} y(0) = c_1 + c_2 &= -10 \\ y'(0) = c_1 + 6c_2 &= 3 \end{align} $$ We solve these equations to get $ c_1= -63/5 $ and $ c_2 = 13/5.$ Finally, we plug our values for $ c_1$ and $ c_2$ into the general solution to find our solution to the initial value problem.

$$
y(x) = -(63/5)\exp(x) + (13/5)\exp(6x)
$$
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©2010, 2014 Andrew G. Bennett