Show that if y(t) satisfies y"-yt=0, then y(-t) satisfies y"+ty=0

[Integrate]

My first instinct was to use the Energy Integral Lemma given from my book
[math]t = \pm \int \frac{1}{\sqrt{2 (F(x) + K)}} dy + c[/math]

K being a constant





Which I feel like is sufficient but the solution manual provides the following.




Which I don't understand and didn't even know that the chain rule could be used on function notation like this.

I guess the s is a stand in for -t.

Crazy that you can use the chain rule like this.


Is what I did sufficient or is the solution provided more correct?

 

[blamocur]

I guess the s is a stand in for -t.

Agree.

Crazy that you can use the chain rule like this.

Why? [imath]s(t) = -t[/imath] is as good a function as any.

Is what I did sufficient or is the solution provided more correct?

I haven't even checked your solution because it is an overkill. Moreover, some differential equations will not have an analytic solutions but the superposition method might still work.