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First Order Linear Equations Additional Examples
Solve the following initial value problem
$$
\begin{align}
\frac{dy}{dx} + 3 y &= -7\exp(9 x) \\
y(0) &= 9
\end{align}
$$
This is a linear equation. First we find the general solution following the paradigm.
Find the integrating factor
$$
\mu(x) = \exp(\int 3 dx) = \exp(3x)
$$
Multiply through by the integrating factor
$$
\exp(3x) \frac{dy}{dx} + 3\exp(3x)y = -7\exp(9 x)\exp(3x) = -7\exp(12x)
$$
Recognize the left-hand-side as $\displaystyle \frac{d}{dx}(\mu(x)y).$
$$
\frac{d}{dx}(\exp(3x)y) =-7\exp(12x)
$$
Integrate both sides.
$$
\exp(3x)y = -(7/12)exp(12x) + C
$$
Divide through by $\mu(x)$ to solve for $ y.$
$$y = -(7/12)exp(9x) + C\exp(-3x)
$$
Now we plug in the initial values $ x = 0 $ and $ y = 9$ and solve for $ C = 115/12$, to obtain the solution to the initial value problem
$$
y = -(7/12)\exp(9x) + (115/12)\exp(-3x)
$$
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