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### First Order Linear Equations

#### Additional Examples

Solve the following initial value problem
$$
\begin{align}
\frac{dy}{dx} - y &= -9\exp(9 x) \\
y(0) &= -8
\end{align}
$$
This is a linear equation. First we find the general solution following the paradigm.
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©2010 Andrew G. Bennett

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- Find the integrating factor $$ \mu(x) = \exp(\int -1 dx) = \exp(-x) $$
- Multiply through by the integrating factor $$ \exp(-x) \frac{dy}{dx} - \exp(-x)y = -9\exp(9 x)\exp(-x) = -9\exp(8x) $$
- Recognize the left-hand-side as $\displaystyle \frac{d}{dx}(\mu(x)y).$ $$ \frac{d}{dx}(\exp(-x)y) =-9\exp(8x) $$
- Integrate both sides. $$ \exp(-x)y = -(9/8)exp(8x) + C $$
- Divide through by $\mu(x)$ to solve for $ y.$ $$y = -(9/8)exp(9x) + C\exp(x) $$

If you have any problems with this page, please contact bennett@math.ksu.edu.

©2010 Andrew G. Bennett