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### Second-Order Linear Inhomogeneous Equations

#### Additional Examples

Solve the initial value problem
$$
\begin{align}
\frac{d^2y}{dx^2} - 4\frac{dy}{dx} + 5y &= \cos(x) \\
y(0) &= -9 \\
y'(0) &= -5
\end{align}
$$

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©2010, 2014 Andrew G. Bennett

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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.*
$$
(D^2 - 4D + 5)y = 0
$$
*Substep 2: Find the roots.*
We use the quadratic formula to compute the roots,
$
2\pm i$

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

Since the right-hand side is $\cos(x),$ 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 $ (4A - 4B)\cos(x) + (4A + 4B)\sin(x) = \cos(x).$ This gives us the equations $4A - 4B=1$ and $4A + 4B=0.$ From these equations we get $ A=1/8$ and $B=-1/8.$ So we get
$$
y_p(x)=(1/8)\cos(x) - (1/8)\sin(x).
$$
__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
$$
y(x) = (1/8)\cos(x) - (1/8)\sin(x) + c_1\exp(2x)\cos(x) + c_2\exp(2x)\sin(x)
$$
**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) = -(1/8)\cos(x) - (1/8)\sin(x) + 2c_1\exp(2x)\cos(x) - c_1\exp(2x)\sin(x) + 2c_2\exp(2x)\sin(x) + c_2\exp(2x)\cos(x).$
Then we plug in $ x=0$ to get the following equations.
$$
\begin{align}
y(0) = 1/8 + c_1 &= -9 \\y'(0) = -1/8 + 2c_1 + c_2 &= -5
\end{align}
$$
We solve these equations to get $ c_1=-73/8$ and $ c_2=107/8.$
Finally, we plug our values for $ c_1$ and $ c_2$ into the general solution to find our solution to the initial value problem.
$$
y(x) = (1/8)\cos(x) - (1/8)\sin(x) - (73/8)\exp(2x)\cos(x) + (107/8)\exp(2x)\sin(x).
$$
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©2010, 2014 Andrew G. Bennett