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### First Order Linear Equations

#### Additional Examples

Solve the following initial value problem
$$
\begin{align}
\frac{dy}{dx} - 9 y &= x + 2 \\
y(0) &= 8
\end{align}
$$
This is a linear equation. First we find the general solution following the paradigm.
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©2010, 2014 Andrew G. Bennett

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- Find the integrating factor $$ \mu(x) = \exp(\int -9 dx) = \exp(-9x) $$
- Multiply through by the integrating factor $$ \exp(-9x) \frac{dy}{dx} - 9\exp(-9x)y = ( x + 2)\exp(-9x) $$
- Recognize the left-hand-side as $\displaystyle \frac{d}{dx}(\mu(x)y).$ $$ \frac{d}{dx}(\exp(-9x)y) =( x + 2)\exp(-9x) $$
- Integrate both sides. In this case you will need to integrate by parts to evaluate the integral on the right.$$ \exp(-9x)y = (-(1/9)x - 19/81)\exp(-9x) + C $$
- Divide through by $\mu(x)$ to solve for $ y.$ $$y = -(1/9)x - 19/81+ C\exp(9x) $$

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

©2010, 2014 Andrew G. Bennett