PX275 - G1 - functions of a single variable

derivative

for a function of one variable, the derivative is defined as:

\frac{d f}{d x} = \frac{f (x + Δ x) - f (x)}{Δ x}

ordinary differential equations (ODEs)

C - first order ODEs
D - second order ODEs
D - second order ODEs

considering an ODE: $\frac{d f}{d x} = a x ⟹ f = a x^{2} + c$
$c$ is a constant of integration, and its value can be determined using an initial condition: $f (0) = c$
a physics example: decay of a radioactive sample:

\frac{d N (t)}{d t} = - λ N (t)

where, $N (t)$ is the number of undecayed nuclei at time, $t$

note:
- if $\frac{d f}{d t} \propto a f (t) ⟹ f \propto e^{a t}$
- similarly, if $\frac{d f}{d t} \propto - a f (t) ⟹ f \propto e^{- a t}$
therefore, the ODE can be solved giving:

N (t) = N_{0} e^{- λ t}

where, $N_{0} = N (t = 0)$

now, considering a second-order ODE:

\frac{d^{2} f}{d x^{2}} \propto a^{2} f

the solution will be exponential
eg: if $f = e^{a x} + e^{- a x} :$

\begin{matrix} \frac{d f}{d x} = a e^{a x} - a e^{- a x} \\ \frac{d^{2} f}{d x^{2}} = a^{2} e^{a x} + a^{2} e^{- a x} = a^{2} f \end{matrix}

considering another second-order ODE:

\frac{d^{2} f}{d x^{2}} \propto - a^{2} f

the solution will be a linear combination of $\cos$ and $\sin$
eg: if $f = \cos a x + \sin a x$

\begin{matrix} \frac{d f}{d x} = - a \sin a x + a \cos a x \\ \frac{d^{2} f}{d x^{2}} = - a^{2} \cos a x - a^{2} \sin a x = - a^{2} f \end{matrix}

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