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. $$ (D^2 - 14D + 98)y = 0 $$ Step 2: Find the roots. We use the quadratic formula to compute the roots, $ 7\pm 7i $

Step 3: Write the general solution.

$$ y(x) = c_1\exp(7x)\cos(7x) + c_2\exp(7x)\sin(7x) $$ 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) = 7c_1\exp(7x)\cos(7x) - 7c_1\exp(7x)\sin(7x) + 7c_2\exp(7x)\sin(7x) + 7c_2\exp(7x)\cos(7x).$ Then we plug in $ x=0 $ to get the following equations. $$ \begin{align} y(0) = c_1 &= -6 \\ y'(0) = 7c_1 + 7c_2 &= -3 \end{align} $$ We solve these equations to get $ c_1= -6 $ and $ c_2 = 39/7.$ Finally, we plug our values for $ c_1$ and $ c_2$ into the general solution to find our solution to the initial value problem.

$$ y(x) = -6\exp(7x)\cos(7x) + (39/7)\exp(7x)\sin(7x) $$ You may reload this page to generate additional examples.