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 $$ \begin{align} \frac{d^2y}{dx^2} - 3\frac{dy}{dx} - 28y &= 0 \\ y(0) &= -10 \\ y'(0) &= 3 \end{align} $$

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. $$ (D^2 - 3D - 28)y = 0 $$ Step 2: Find the roots. The equation factors into $(D + 4)(D - 7)y = 0.$ So our roots are -4 and 7.

Step 3: Write the general solution.

$$ y(x) = c_1\exp(-4x) + c_2\exp(7x) $$ 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) = -4c_1\exp(-4x) + 7c_2\exp(7x).$ Then we plug in $ x=0 $ to get the following equations. $$ \begin{align} y(0) = c_1 + c_2 &= -10 \\ y'(0) = -4c_1 + 7c_2 &= 3 \end{align} $$ We solve these equations to get $ c_1= -73/11 $ and $ c_2 = -37/11.$ Finally, we plug our values for $ c_1$ and $ c_2$ into the general solution to find our solution to the initial value problem.

$$ y(x) = -(73/11)\exp(-4x) - (37/11)\exp(7x) $$ 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