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First Order Linear Equations Additional Examples
Solve the following initial value problem
$$
\begin{align}
\frac{dy}{dx} - 8 y &= 6\exp(-6 x) \\
y(0) &= -6
\end{align}
$$
This is a linear equation. First we find the general solution following the paradigm.
Find the integrating factor
$$
\mu(x) = \exp(\int -8 dx) = \exp(-8x)
$$
Multiply through by the integrating factor
$$
\exp(-8x) \frac{dy}{dx} - 8\exp(-8x)y = 6\exp(-6 x)\exp(-8x) = 6\exp(-14x)
$$
Recognize the left-hand-side as $\displaystyle \frac{d}{dx}(\mu(x)y).$
$$
\frac{d}{dx}(\exp(-8x)y) =6\exp(-14x)
$$
Integrate both sides.
$$
\exp(-8x)y = -(3/7)exp(-14x) + C
$$
Divide through by $\mu(x)$ to solve for $ y.$
$$y = -(3/7)exp(-6x) + C\exp(8x)
$$
Now we plug in the initial values $ x = 0 $ and $ y = -6$ and solve for $ C = -39/7$, to obtain the solution to the initial value problem
$$
y = -(3/7)\exp(-6x) - (39/7)\exp(8x)
$$
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