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### First Order Linear Equations

#### Additional Examples

Solve the following initial value problem
$$
\begin{align}
\frac{dy}{dx} + 2 y &= -9 x \\
y(0) &= 10
\end{align}
$$
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 $$ \mu(x) = \exp(\int 2 dx) = \exp(2x) $$
- Multiply through by the integrating factor $$ \exp(2x) \frac{dy}{dx} + 2\exp(2x)y = (-9 x)\exp(2x) $$
- Recognize the left-hand-side as $\displaystyle \frac{d}{dx}(\mu(x)y).$ $$ \frac{d}{dx}(\exp(2x)y) =(-9 x)\exp(2x) $$
- Integrate both sides. In this case you will need to integrate by parts to evaluate the integral on the right.$$ \exp(2x)y = (-(9/2)x + 9/4)\exp(2x) + C $$
- Divide through by $\mu(x)$ to solve for $ y.$ $$y = -(9/2)x + 9/4+ C\exp(-2x) $$

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

©2010 Andrew G. Bennett