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. $$ (D^2 + 9D + 8)y = 0 $$ Step 2: Find the roots. The equation factors into $(D + 8)(D + 1)y = 0.$ So our roots are -8 and -1.

Step 3: Write the general solution.

$$ y(x) = c_1\exp(-8x) + c_2\exp(-x) $$ 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) = -8c_1\exp(-8x) - c_2\exp(-x).$ Then we plug in $ x=0 $ to get the following equations. $$ \begin{align} y(0) = c_1 + c_2 &= 7 \\ y'(0) = -8c_1 - c_2 &= -1 \end{align} $$ We solve these equations to get $ c_1= -6/7 $ and $ c_2 = 55/7.$ Finally, we plug our values for $ c_1$ and $ c_2$ into the general solution to find our solution to the initial value problem.

$$ y(x) = -(6/7)\exp(-8x) + (55/7)\exp(-x) $$ You may reload this page to generate additional examples.