Differential equations

Differential Equation

For any differential equation has the general form

${\frac {d}{dt}}f(t)=f(t)$

Solving differential equation

${\frac {df(t)}{f(t)}}=dt$
$\int {\frac {df(t)}{f(t)}}=\int dt$
$Lnf(t)=t+c$
$f(t)=e^{t+c}=Ae^{t}$  with $A=e^{c}$

Similarly,

${\frac {d}{dt}}f(t)=sf(t)$
$f(t)=e^{t+c}=Ae^{st}$  with $A=e^{c}$

In general,

 Equation General Form Root of equation Differential equation ${\frac {d}{dt}}f(t)=f(t)$ $f(t)=e^{t+c}=Ae^{t}$ Differential equation ${\frac {d}{dt}}f(t)=-f(t)$ $f(t)=e^{-t+c}=Ae^{-t}$ Differential equation ${\frac {d}{dt}}f(t)=sf(t)$ $f(t)=e^{st+c}=Ae^{st}$ Differential equation ${\frac {d}{dt}}f(t)=-sf(t)$ $f(t)=e^{-st+c}=Ae^{-st}$ Ordered differential equations

 Equation General Form Root of equation 1st order differential equation ${\frac {d}{dt}}f(t)=-sf(t)$ $f(t)=e^{-st+c}=Ae^{-st}$ 2nd order differential equation ${\frac {d^{2}}{dt^{2}}}f(t)=-sf(t)$ $f(t)=e^{\pm j{\sqrt {s}}t+c}=Ae^{\pm j\omega t}=ASin\omega t$ . With, $\omega ={\sqrt {s}}$ nth order differential equation ${\frac {d^{n}}{dt^{n}}}f(t)=-sf(t)$ $f(t)=e^{\pm jn{\sqrt {s}}t+c}=Ae^{\pm j\omega t}=ASin\omega t$ . With, $\omega =n{\sqrt {s}}$ Ordinary differential equation

1st order ordinary differential equation

1st order ordinary differential equation of general form

$A{\frac {d}{dt}}f(x)+Bf(t)=0$

Rearrange equation above,

${\frac {d}{dt}}f(t)=-sf(t)$  . With, $s={\frac {B}{A}}$

Root of equation

$f(t)=Ae^{-st}=Ae^{-{\frac {B}{A}}t}$

2nd order ordinary differential equation

$A{\frac {d^{2}}{dx^{2}}}f(x)+B{\frac {d}{dx}}f(x)+Cf(x)=0$
${\frac {d^{2}}{dx^{2}}}f(x)+{\frac {B}{A}}{\frac {d}{dx}}f(x)+{\frac {C}{A}}f(x)=0$
${\frac {d^{2}}{dx^{2}}}f(x)=-{\frac {B}{A}}{\frac {d}{dx}}f(x)-{\frac {C}{A}}f(x)$
${\frac {d^{2}}{dx^{2}}}f(x)=-2\alpha {\frac {d}{dx}}f(x)-\beta f(x)$
$\beta ={\frac {C}{A}}$
$\alpha =\beta \gamma ={\frac {C}{A}}\gamma ={\frac {B}{2A}}$
$\gamma ={\frac {B}{2C}}$

Roots of differential equations

• 1 real root . $f(x)=Ae^{-\alpha t}$  . $\alpha =\beta$
• 2 real roots . $f(x)=Ae^{(-\alpha \pm {\sqrt {\alpha -\beta }})t}$  . $\alpha >\beta$
• 2 complex roots . $f(x)=Ae^{(-\alpha \pm {\sqrt {\beta -\alpha }})t}$  . $\alpha <\beta$
$f(x)=Ae^{(-\alpha \pm {\sqrt {\beta -\alpha }})t}=A(\alpha )Sin\omega t$

With

$A(\alpha )=Ae^{-\alpha t}$
$\omega ={\sqrt {\beta -\alpha }}$

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