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Tuesday, June 4, 2024

Deriving the Solution to First-Order Linear Differential Equations

\newcommand{\dif}{\mathrm{d}} 
A first-order linear differential equation is one of many irritating nuisances encountered in calculus. I thus present to you its rancid definition:
    A first-order linear diffential equation is any differential equation of the form 
\frac{\dif x}{\dif t} + p(t)x = q(t)

Which has the plain and simple solution 
x = e^{-\int p(t) \dif t} \int e^{\int p(t) \dif t}q(t)\dif t
If you too are confused at this, then you've come to the right place.

First, notice that a first-order linear is very similar to the form \frac{\dif u}{\dif t}v+u\frac{\dif v}{\dif t}, which by the product rule would simply equal \frac{\dif }{\dif t}(uv). Our goal is therefore to put the DE into this form.

The key step is to invent a new function, called f(t), and multiply the first-order linear DE by it:
\frac{\dif x}{\dif t}f(t) + f(t)p(t)x = f(t)q(t)
And if we define f(t)p(t) equal to \frac{\dif f}{\dif t}, the whole thing can simplify with the product rule, just as we expected: 
\frac{\dif x}{\dif t}f(t) + \frac{\dif f}{\dif t}x = f(t)q(t)
\frac{\dif }{\dif t}(x \cdot f(t)) = f(t)q(t)

Which is much more manageable. 

x \cdot f(t) = \int f(t)q(t) \dif t

x = \frac{1}{f(t)} \int f(t)q(t) \dif t

And you have the solution, assuming you're fine with not explicitly defining f. If that is not the case, then let's find f.

The only condition we gave was

\frac{\dif f}{\dif t}=f(t)p(t)

Rearanging and integrating:

\int  \frac{1}{f(t)}\dif f = \int p(t) \dif t 

\ln(f(t)) = \int p(t) \dif t

f(t) = e^{\int p(t) \dif t}

And finally substituting it into the solution:

x = e^{-\int p(t) \dif t} \int e^{\int p(t) \dif t}q(t)\dif t

\mathbb{Q.E.D.}

Deriving the Solution to First-Order Linear Differential Equations

\newcommand{\dif}{\mathrm{d}}  A first-order linear differential equation is one of many irritating nuisances encountered in calculus. I...