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Second-Order Linear Homogeneous Equations

Additional Examples

Solve the initial value problem $$ \begin{align} \frac{d^2y}{dx^2} + 8\frac{dy}{dx} + 97y &= 0 \\ y(0) &= 5 \\ y'(0) &= -3 \end{align} $$

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. $$ (D^2 + 8D + 97)y = 0 $$ Step 2: Find the roots. We use the quadratic formula to compute the roots, $ -4\pm 9i $

Step 3: Write the general solution.

$$ y(x) = c_1\exp(-4x)\cos(9x) + c_2\exp(-4x)\sin(9x) $$ 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(9x) - 9c_1\exp(-4x)\sin(9x) - 4c_2\exp(-4x)\sin(9x) + 9c_2\exp(-4x)\cos(9x).$ Then we plug in $ x=0 $ to get the following equations. $$ \begin{align} y(0) = c_1 &= 5 \\ y'(0) = -4c_1 + 9c_2 &= -3 \end{align} $$ We solve these equations to get $ c_1= 5 $ and $ c_2 = 17/9.$ Finally, we plug our values for $ c_1$ and $ c_2$ into the general solution to find our solution to the initial value problem.

$$ y(x) = 5\exp(-4x)\cos(9x) + (17/9)\exp(-4x)\sin(9x) $$ You may reload this page to generate additional examples.


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