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. $$ (D^2 + 10D + 41)y = 0 $$ Step 2: Find the roots. We use the quadratic formula to compute the roots, $ -5\pm 4i $

Step 3: Write the general solution.

$$ y(x) = c_1\exp(-5x)\cos(4x) + c_2\exp(-5x)\sin(4x) $$ 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) = -5c_1\exp(-5x)\cos(4x) - 4c_1\exp(-5x)\sin(4x) - 5c_2\exp(-5x)\sin(4x) + 4c_2\exp(-5x)\cos(4x).$ Then we plug in $ x=0 $ to get the following equations. $$ \begin{align} y(0) = c_1 &= 5 \\ y'(0) = -5c_1 + 4c_2 &= -3 \end{align} $$ We solve these equations to get $ c_1= 5 $ and $ c_2 = 11/2.$ Finally, we plug our values for $ c_1$ and $ c_2$ into the general solution to find our solution to the initial value problem.

$$ y(x) = 5\exp(-5x)\cos(4x) + (11/2)\exp(-5x)\sin(4x) $$ You may reload this page to generate additional examples.