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Second-Order Linear Homogeneous Equations

Solve the initial value problem \begin{align} \frac{d^2y}{dx^2} - 10\frac{dy}{dx} + 50y &= 0 \\ y(0) &= -6 \\ y'(0) &= 9 \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. $$(D^2 - 10D + 50)y = 0$$ Step 2: Find the roots. We use the quadratic formula to compute the roots, $5\pm 5i$

Step 3: Write the general solution.

$$y(x) = c_1\exp(5x)\cos(5x) + c_2\exp(5x)\sin(5x)$$ 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, $y'(x) = 5c_1\exp(5x)\cos(5x) - 5c_1\exp(5x)\sin(5x) + 5c_2\exp(5x)\sin(5x) + 5c_2\exp(5x)\cos(5x).$ Then we plug in $x=0$ to get the following equations. \begin{align} y(0) = c_1 &= -6 \\ y'(0) = 5c_1 + 5c_2 &= 9 \end{align} We solve these equations to get $c_1= -6$ and $c_2 = 39/5.$ Finally, we plug our values for $c_1$ and $c_2$ into the general solution to find our solution to the initial value problem.

$$y(x) = -6\exp(5x)\cos(5x) + (39/5)\exp(5x)\sin(5x)$$ You may reload this page to generate additional examples.