Differential equations

Differential Equation

For any differential equation has the general form

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

Solving differential equation

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

Similarly,

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

In general,

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

Ordered differential equations

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

Ordinary differential equation

1st order ordinary differential equation

1st order ordinary differential equation of general form

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

Rearrange equation above,

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

Root of equation

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

2nd order ordinary differential equation

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

Roots of differential equations

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

With

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

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