Not including four play in games, there are 64 teams in the tournament. once a team loses one game they are eliminated. how many
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First, I'll make f(x) = sin(px) + cos(px) because this expression shows up quite a lot, and such a substitution makes life a bit easier for us.
Let's apply the first derivative of this f(x) function.
Now apply the derivative to that to get the second derivative
We can see that f '' (x) is just a scalar multiple of f(x). That multiple of course being -p^2.
Keep in mind that we haven't actually found dy/dx yet, or its second derivative counterpart either.
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Let's compute dy/dx. We'll use f(x) as defined earlier.
Use the chain rule here.
There's no need to plug in the expressions f(x) or f ' (x) as you'll see in the last section below.
Now use the quotient rule to find the second derivative of y
If you need a refresher on the quotient rule, then
where P and Q are functions of x.
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This then means
Note the cancellation of -(f ' (x))^2 with (f ' (x))^2
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Let's then replace f '' (x) with -p^2*f(x)
This allows us to form ( f(x) )^2 in the numerator to cancel out with the denominator.
So this concludes the proof that when
Side note: This is an example of showing that the given y function is a solution to the given second order linear differential equation.