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### Second-Order Linear Inhomogeneous Equations

Solve the initial value problem \begin{align} \frac{d^2y}{dx^2} - 6\frac{dy}{dx} + 18y &= 3\sin(x) \\ y(0) &= -5 \\ y'(0) &= 8 \end{align}

This is a second-order linear constant-coefficient initial value problem.

First, we find the general solution

Step 1: Find the homogeneous solution.

Substep 1: Write the equation in operator form. $$(D^2 - 6D + 18)y = 0$$ Substep 2: Find the roots. We use the quadratic formula to compute the roots, $3\pm 3i$

Substep 3: Write the homogeneous solutions. $$y_h(x) = c_1\exp(3x)\cos(3x) + c_2\exp(3x)\sin(3x)$$ Step 2: Guess the form of the particular solutions

Since the right-hand side is $3\sin(x),$ we guess $y_p(x) = A\cos(x) + B\sin(x).$

Step 3: Plug our guess into the equation and solve for the undetermined coefficients

Plugging our guess for $y_p(x)$ into the equation, we obtain $(17A - 6B)\cos(x) + (6A + 17B)\sin(x) = 3\sin(x).$ This gives us the equations $17A - 6B=0$ and $6A + 17B=3.$ From these equations we get $A=18/325$ and $B=51/325.$ So we get $$y_p(x)=(18/325)\cos(x) + (51/325)\sin(x).$$ Step 4: The general solution is the particular solution plus all the homogeneous solutions.

So from the results of steps 1 and 3, we get the general solution is $$y(x) = (18/325)\cos(x) + (51/325)\sin(x) + c_1\exp(3x)\cos(3x) + c_2\exp(3x)\sin(3x)$$ Second, we solve for the constants

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, $y'(x) = (51/325)\cos(x) - (18/325)\sin(x) + 3c_1\exp(3x)\cos(3x) - 3c_1\exp(3x)\sin(3x) + 3c_2\exp(3x)\sin(3x) + 3c_2\exp(3x)\cos(3x).$ Then we plug in $x=0$ to get the following equations. \begin{align} y(0) = 18/325 + c_1 &= -5 \\y'(0) = 51/325 + 3c_1 + 3c_2 &= 8 \end{align} We solve these equations to get $c_1=-1643/325$ and $c_2=7478/975.$ Finally, we plug our values for $c_1$ and $c_2$ into the general solution to find our solution to the initial value problem. $$y(x) = (18/325)\cos(x) + (51/325)\sin(x) - (1643/325)\exp(3x)\cos(3x) + (7478/975)\exp(3x)\sin(3x).$$ You may reload this page to generate additional examples.