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### Second-Order Linear Homogeneous Equations

Solve the initial value problem \begin{align} \frac{d^2y}{dx^2} - 18\frac{dy}{dx} + 81y &= 0 \\ y(0) &= 6 \\ y'(0) &= -4 \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. $$(D^2 - 18D + 81)y = 0$$ Step 2: Find the roots. The equation factors into $(D - 9)^2 y = 0,$ so we have a double root of 9

Step 3: Write the general solution.

$$y(x) = c_1\exp(9x) + c_2x\exp(9x)$$ 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, $y'(x) = 9c_1\exp(9x) + c_2\exp(9x) + 9c_2x\exp(9x).$ Then we plug in $x=0$ to get the following equations. \begin{align} y(0) = c_1 &= 6 \\ y'(0) = 9c_1 + c_2 &= -4 \end{align} We solve these equations to get $c_1= 6$ and $c_2 = -58.$ Finally, we plug our values for $c_1$ and $c_2$ into the general solution to find our solution to the initial value problem.

$$y(x) = 6\exp(9x) - 58x\exp(9)$$ You may reload this page to generate additional examples.