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

Step 3: Write the general solution.

$$ y(x) = c_1\exp(7x) + c_2\exp(-4x) $$ 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) = 7c_1\exp(7x) - 4c_2\exp(-4x).$ Then we plug in $ x=0 $ to get the following equations. $$ \begin{align} y(0) = c_1 + c_2 &= 1 \\ y'(0) = 7c_1 - 4c_2 &= 2 \end{align} $$ We solve these equations to get $ c_1= 6/11 $ and $ c_2 = 5/11.$ Finally, we plug our values for $ c_1$ and $ c_2$ into the general solution to find our solution to the initial value problem.

$$ y(x) = (6/11)\exp(7x) + (5/11)\exp(-4x) $$ You may reload this page to generate additional examples.