Question

Gleichung lösen:

-50e^{-0.5t} + 100e^{-t} = 0

Wie muss ich rechnen?

Answer 1

-50·e^{-0.5*t} +100·e^{-t} = 0  | +100·e^{-t} | :(-50)

e^{-0.5*t} = 2*e^{-t}  | ln ( )
-0.5*t = ln (  2*e^{-t}  ) = ln ( 2 ) + ln ( e^{-t} )
-0.5*t = ln ( 2 ) + (-t)  = ln (2) - t
0.5 * t = ln ( 2 )
t = ln (2 ) / 0.5 = 1.386

Probe
-50*e^{-0.5*t}  + 100 * e^{-t} = 0
-50*e^{-0.5*1.386}  + 100 * e^{-1.386} = 0
-50 * 0.5 + 100 *  0.25 =  0  | stimmt

Answer 2

-50e^{-0.5t} + 100e^{-t} = 0

substituiere u = e^{-0.5t}, wobei u>0.

-50u + 100u^2 = 0

50u( - 1 + 2u) = 0

1. Lösung u1 = 0 ergibt kein passendes t

2. Lösung u2 = 1/2

Rücksubstituition:

1/2 = e^{-0.5t}       | Kehrwert

2 = e^{0.5t}          |ln

ln(2) = 0.5t      |*2

2* ln(2) = t