Help with q25 please. Thanks.​

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.

its 4 because 2 times 2 is 4, if you double the number two you get four

. + λrh() e 1 1 2 4 4 4 4 4 3 4 4 4 4 4 + th−1(− s(x)2 dx ω + λrh−1() e 2 1 2 4 4

step-by-step explanation: