Kansas State University

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. $$ (D^2 - 8D + 20)y = 0 $$ Step 2: Find the roots. We use the quadratic formula to compute the roots, $ 4\pm 2i $

Step 3: Write the general solution.

$$ y(x) = c_1\exp(4x)\cos(2x) + c_2\exp(4x)\sin(2x) $$ 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)\cos(2x) - 2c_1\exp(4x)\sin(2x) + 4c_2\exp(4x)\sin(2x) + 2c_2\exp(4x)\cos(2x).$ Then we plug in $ x=0 $ to get the following equations. $$ \begin{align} y(0) = c_1 &= 3 \\ y'(0) = 4c_1 + 2c_2 &= -1 \end{align} $$ We solve these equations to get $ c_1= 3 $ and $ c_2 = -13/2.$ Finally, we plug our values for $ c_1$ and $ c_2$ into the general solution to find our solution to the initial value problem.

$$ y(x) = 3\exp(4x)\cos(2x) - (13/2)\exp(4x)\sin(2x) $$ You may reload this page to generate additional examples.