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Second-Order Linear Homogeneous Equations

Additional Examples

Solve the initial value problem $$ \begin{align} \frac{d^2y}{dx^2} + \frac{dy}{dx} - 72y &= 0 \\ y(0) &= -9 \\ y'(0) &= 8 \end{align} $$

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. $$ (D^2 + D - 72)y = 0 $$ Step 2: Find the roots. The equation factors into $(D - 8)(D + 9)y = 0.$ So our roots are 8 and -9.

Step 3: Write the general solution.

$$ y(x) = c_1\exp(8x) + c_2\exp(-9x) $$ 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) = 8c_1\exp(8x) - 9c_2\exp(-9x).$ Then we plug in $ x=0 $ to get the following equations. $$ \begin{align} y(0) = c_1 + c_2 &= -9 \\ y'(0) = 8c_1 - 9c_2 &= 8 \end{align} $$ We solve these equations to get $ c_1= -73/17 $ and $ c_2 = -80/17.$ Finally, we plug our values for $ c_1$ and $ c_2$ into the general solution to find our solution to the initial value problem.

$$ y(x) = -(73/17)\exp(8x) - (80/17)\exp(-9x) $$ You may reload this page to generate additional examples.


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