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### Second-Order Linear Inhomogeneous Equations

#### Additional Examples

Solve the initial value problem
$$
\begin{align}
\frac{d^2y}{dx^2} - 3\frac{dy}{dx} - 40y &= -6x - 5 \\
y(0) &= 7 \\
y'(0) &= 9
\end{align}
$$

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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.*
$$
(D^2 - 3D - 40)y = 0
$$
*Substep 2: Find the roots.*
The equation factors into $ (D - 8)(D + 5)y = 0.$
So our roots are 8 and -5.

*Substep 3: Write the homogeneous solutions.*
$$
y_h(x) = c_1\exp(8x) + c_2\exp(-5x)
$$
__Step 2: Guess the form of the particular solutions__

Since the right-hand side is $-6x - 5,$ we guess $ y_p(x) = Ax + B.$

__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 $ -40Ax - 40B - 3A = -6x - 5.$ This gives us the equations $-40A=-6$ and $-40B - 3A=-5.$ From these equations we get $A=3/20$ and $B=91/800.$ So we get
$$
y_p(x)=(3/20)x + 91/800.
$$
__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
$$
y(x) = (3/20)x + 91/800 + c_1\exp(8x) + c_2\exp(-5x)
$$
**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) = 3/20 + 8c_1\exp(8x) - 5c_2\exp(-5x).$
Then we plug in $ x=0$ to get the following equations.
$$
\begin{align}
y(0) = 91/800 + c_1 + c_2 &= 7 \\y'(0) = 3/20 + 8c_1 - 5c_2 &= 9
\end{align}
$$
We solve these equations to get $ c_1=1385/416$ and $ c_2=1156/325.$
Finally, we plug our values for $ c_1$ and $ c_2$ into the general solution to find our solution to the initial value problem.
$$
y(x) = (3/20)x + 91/800 + (1385/416)\exp(8x) + (1156/325)\exp(-5x).
$$
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©2010, 2014 Andrew G. Bennett