SGR
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What function equals the negative derivative of itself?
f(x) = -f'(x)
f(x) = -f'(x)
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SGR said:What equation equals the negative derivative of its self?
f(x) = f'(x)
SGR said:YES! when does f(x) = -f'(x)??
MarkFL said:Let's use the notation of Leibniz and write:
$$\d{f}{x}=-f(x)$$
Now what if we separate variables and write:
$$\frac{1}{f(x)}\,df=-dx$$
We have divided by $f(x)$, and in doing so potentially eliminated the trivial solution $f(x)=0$, and so we need to be mindful of this in case the non-trivial solution does not include this.
What do you get when you integrate both sides of the above equation?
SGR said:I don't know how to integrate.
MarkFL said:Let's use the notation of Leibniz and write:
$$\d{f}{x}=-f(x)$$
Now what if we separate variables and write:
$$\frac{1}{f(x)}\,df=-dx$$
We have divided by $f(x)$, and in doing so potentially eliminated the trivial solution $f(x)=0$, and so we need to be mindful of this in case the non-trivial solution does not include this.
What do you get when you integrate both sides of the above equation?