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
$$
\begin{align}
\frac{d^2y}{dx^2} - 16\frac{dy}{dx} + 100y &= 0 \\
y(0) &= 3 \\
y'(0) &= -10
\end{align}
$$

*
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.*
$$
(D^2 - 16D + 100)y = 0
$$
*Step 2: Find the roots.*
We use the quadratic formula to compute the roots,
$ 8\pm 6i $

*Step 3: Write the general solution.*

$$ y(x) = c_1\exp(8x)\cos(6x) + c_2\exp(8x)\sin(6x) $$ 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) = 8c_1\exp(8x)\cos(6x) - 6c_1\exp(8x)\sin(6x) + 8c_2\exp(8x)\sin(6x) + 6c_2\exp(8x)\cos(6x).$ Then we plug in $ x=0 $ to get the following equations. $$ \begin{align} y(0) = c_1 &= 3 \\ y'(0) = 8c_1 + 6c_2 &= -10 \end{align} $$ We solve these equations to get $ c_1= 3 $ and $ c_2 = -17/3.$ Finally, we plug our values for $ c_1$ and $ c_2$ into the general solution to find our solution to the initial value problem.

$$
y(x) = 3\exp(8x)\cos(6x) - (17/3)\exp(8x)\sin(6x)
$$
*You may reload this page to generate additional examples.
*

*
*

©2010, 2014 Andrew G. Bennett