Question 1184075: Suppose x=e^(2t).
a)Find the value of the expression x′′′−6x′′+12x′−8x in terms of the variable t. (Enter the terms in the order given.)
My answer is 8e^(2t)-24e^(2t)+24e^(2t)-8e^(2t), correct.
b)Simplify your answer to the previous part and enter a differential equation in terms of the dependent variable x satisfied by x=e^(2t). Enter the derivatives of x using prime notation (x′,x′′,x′′′).
My answer is 8e^(2t)-24e^(2t)+24e^(2t)-8e^(2t)=0 , wrong
Answer by CPhill(1987)   (Show Source):
You can put this solution on YOUR website! **a) Finding the expression in terms of *t***
You're absolutely correct! Here's how we get there:
* **x = e^(2t)**
* **x' = 2e^(2t)** (First derivative)
* **x'' = 4e^(2t)** (Second derivative)
* **x''' = 8e^(2t)** (Third derivative)
Now, substitute these into the expression x''' - 6x'' + 12x' - 8x:
8e^(2t) - 6(4e^(2t)) + 12(2e^(2t)) - 8(e^(2t)) = 8e^(2t) - 24e^(2t) + 24e^(2t) - 8e^(2t)
**b) Simplifying and the differential equation**
As you've already noticed, the expression simplifies to 0:
8e^(2t) - 24e^(2t) + 24e^(2t) - 8e^(2t) = 0
Therefore, the differential equation satisfied by x = e^(2t) is:
x''' - 6x'' + 12x' - 8x = 0
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