Question

suppose f(x) satisfies the following equation f''(x)=-f(x). show that (f(x))^2+(f'(x))^2=constant for all x

Answer 1

\[f''(x)=-f(x)\]

show that

\[(f(x))^2+(f'(x))^2=constant\]

Answer 2

@SithsAndGiggles

Answer 3

this question is evil

Answer 4

Have you taken differential equations yet? That'd probably be the easiest way to figure this one out.

Answer 5

no, this is for calc 1

Answer 6

i doubt this is gonna be on the final that i have in an hour

Answer 7

we didnt go over it in class but its on the review

Answer 8

Maybe it's a matter of pattern recognition, in which case you should recall that \(\dfrac{d}{dx}\sin x=\cos x\) and \(\dfrac{d^2}{dx}\sin x=-\sin x\), which satisfies the condition \(f''(x)=-f(x)\). The same applies for \(f(x)=\cos x\).

Answer 9

As well as for \(f(x)=\sin x+\cos x\).

Answer 10

so the constant is 1?

Answer 11

ya interesting

Answer 12

In this case, yes, but you can see that the condition is met for all \(k\) with \(f(x)=k(\sin x+\cos x)\).

Answer 13

The question is how to arrive at these functions for \(f\). The DE-way is easy enough, but probably beyond what you've learned if you haven't been exposed to differential equations yet.

Answer 14

it seems like a pretty unique question, so at least i know the answer, thanks