Math 240 Home,
Textbook Contents,
Online Homework Home

**Warning: MathJax
requires JavaScript to process the mathematics on this page.**

If your browser supports JavaScript, be sure it is enabled.**
****
**

### Second-Order Linear Homogeneous Equations

#### Additional Examples

Solve the initial value problem
$$
\begin{align}
\frac{d^2y}{dx^2} + 10\frac{dy}{dx} + 9y &= 0 \\
y(0) &= 7 \\
y'(0) &= -1
\end{align}
$$

*
If you have any problems with this page, please contact
bennett@math.ksu.edu.*

©2010, 2014 Andrew G. Bennett

If your browser supports JavaScript, be sure it is enabled.

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.*
$$
(D^2 + 10D + 9)y = 0
$$
*Step 2: Find the roots.*
The equation factors into $(D + 1)(D + 9)y = 0.$
So our roots are -1 and -9.

*Step 3: Write the general solution.*

$$ y(x) = c_1\exp(-x) + c_2\exp(-9x) $$ 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) - 9c_2\exp(-9x).$ Then we plug in $ x=0 $ to get the following equations. $$ \begin{align} y(0) = c_1 + c_2 &= 7 \\ y'(0) = -c_1 - 9c_2 &= -1 \end{align} $$ We solve these equations to get $ c_1= 31/4 $ and $ c_2 = -3/4.$ Finally, we plug our values for $ c_1$ and $ c_2$ into the general solution to find our solution to the initial value problem.

$$
y(x) = (31/4)\exp(-x) - (3/4)\exp(-9x)
$$
*You may reload this page to generate additional examples.
*

*
*

©2010, 2014 Andrew G. Bennett