Section C.3 Integrating Factor
Standard Form.
Every linear differential equation looks like
\begin{equation*}
a_n(x)y^{(n)} + a_{n-1}(x)y^{(n-1)} + \cdots + a_2(x)y'' + a_1(x)y' + a_0(x)y = f(x)\text{.}
\end{equation*}
The first-order version of this is
\begin{equation*}
a_1(x)y' + a_0(x)y = f(x)\text{,}
\end{equation*}
but we can divide both sides of this equation by \(a_1(x)\text{,}\) like so
\begin{equation*}
y' + \us{P(x)}{\ub{\frac{a_0(x)}{a_1(x)}}}y = \us{Q(x)}{\ub{\frac{f(x)}{a_1(x)}}}\text{.}
\end{equation*}
Since every first-order linear differential equations can always be written as
\begin{equation*}
y' + P(x)y = Q(x) \text{,}
\end{equation*}
we call this the first-order linear standard form.Integrating Factor Calculation Details.
\begin{align*}
\frac{d\mu}{dx} =\amp\ 2\mu \\
\frac{1}{\mu}d\mu =\amp\ 2 dx \\
\int \frac{1}{\mu}d\mu =\amp\ \int 2 dx \\
\ln|\mu| =\amp\ 2x + c \quad (c = 0)\\
\mu =\amp\ e^{2x}
\end{align*}
Since we only need one integrating factor, we are free to pick any value of \(c\text{.}\)You have attempted of activities on this page.
