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Suppose $a$ and $b$ are expressions in terms of the variable $x$.

We know:

$\begin{align} a &= a \cdot\frac{b}{b} \\ &= \frac{ab}{b} \\ \end{align} $

Is there a systematic way to find $b$ such that:

$\frac{db}{dx} = ab$

For example:

If $a=\sec(x)$, then $b=\tan(x)+sec(x)$ since $\frac{db}{dx} = sec(x)tan(x)+sec^2(x)=ab$

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  • $\begingroup$ If you have $a$ given and want to find $b$, then this is called an ODE (ordinary differential equation). Solving it may become quite difficult depending on $a$. $\endgroup$
    – AlexR
    Nov 29, 2013 at 12:43

3 Answers 3

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Yes! In fact, this is how to solve first order linear differential equations:

$y'(x)+a(x)y(x)=f(x)$,

by multiplying both sides by b(x). Then if $b'(x)=a(x)b(x)$, we have

$\frac{d}{dx}(b(x)y(x))=f(x)b(x)$,

which can be solved for $y(x)$ by integrating.

To find the `integrating factor' $b(x)$, note that we can write the condition as

$\frac{d}{dx}\log(b(x))=a(x)$

which can be systematically solved by integrating.

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With this specific form the solution is not too difficult. Note that the derivative of $\ln b(x)$ is $\frac{b'(x)}{b(x)}$ and that should equal $a(x)$. So $$ b(x) = e^{\int a(x)\,\mathrm dx}.$$ Note that the additive integration constant for the integral becomes a multiplicative constant for $b$.

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As AlexR says, its a differential equation. The solution of it can be found by: $$ \frac{db}{b} = a(x)\,dx\quad \Longrightarrow \quad \ln(b) =\int a(x)\,dx \quad \Longrightarrow \quad b(x) = e^{\int a(x)\,dx}$$

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