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 $$ \begin{align} \frac{d^2y}{dx^2} + 4\frac{dy}{dx} - 32y &= 0 \\ y(0) &= -8 \\ y'(0) &= 6 \end{align} $$

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. $$ (D^2 + 4D - 32)y = 0 $$ Step 2: Find the roots. The equation factors into $(D + 8)(D - 4)y = 0.$ So our roots are -8 and 4.

Step 3: Write the general solution.

$$ y(x) = c_1\exp(-8x) + c_2\exp(4x) $$ 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) + 4c_2\exp(4x).$ Then we plug in $ x=0 $ to get the following equations. $$ \begin{align} y(0) = c_1 + c_2 &= -8 \\ y'(0) = -8c_1 + 4c_2 &= 6 \end{align} $$ We solve these equations to get $ c_1= -19/6 $ and $ c_2 = -29/6.$ Finally, we plug our values for $ c_1$ and $ c_2$ into the general solution to find our solution to the initial value problem.

$$ y(x) = -(19/6)\exp(-8x) - (29/6)\exp(4x) $$ You may reload this page to generate additional examples.


If you have any problems with this page, please contact bennett@math.ksu.edu.
©2010, 2014 Andrew G. Bennett