- #1
Juanriq
- 42
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Homework Statement
Reduce the order of a Cauchy-Euler EquationHomework Equations
[itex] x = e^t \mbox{ and } \ln x = t [/itex]The Attempt at a Solution
[itex] \displaystyle \frac{d y}{d x} = \displaystyle \frac{d y}{d t} \displaystyle \frac{d t}{d x} = \displaystyle \frac{d y}{d t} \cdot \displaystyle \frac{1}{x}
[/itex]
and thus
[itex]
\displaystyle \frac{d^2 y}{d x^2} = \displaystyle \frac{d y}{d t} \cdot \displaystyle \frac{-1}{x^2} + \displaystyle \frac{1}{x} \displaystyle \frac{d}{d x} \Bigl ( \displaystyle \frac{d y}{d t} \Bigl )
[/itex]
Here is where I am getting stuck, specifically on [itex] \displaystyle \frac{d}{d x} \Bigl ( \displaystyle \frac{d y}{d t} \Bigl )
[/itex] this step. I know what I should get...
[itex]
\displaystyle \frac{1}{x} \Bigl ( \displaystyle \frac{d^2 y}{d t^2} \cdot \displaystyle \frac{1}{x} \Bigl )
[/itex]
But uhhh not getting it. Thanks in advance!