Im trying to understand a "simple" identity.(adsbygoogle = window.adsbygoogle || []).push({});

I have the problem [tex]\ddot{y}+g(t,y)=0[/tex]

with [tex]y(0)=y_{0}[/tex] and [tex]\dot{y}(0)=z_{0}[/tex]

What i need to prove is that the solution can be expresed in the form

[tex]y(t)=y_{0}+tz_{0}-\int_{0}^{t}(t-s)g(s,y(s))ds[/tex]

Integrating twice is clear that

[tex]y(t)=y_{0}+tz_{0}-\int_{0}^{t}\{\int_{0}^{s}g(\tau,y(\tau))d\tau\}ds[/tex]

Now the book goes

[tex]\int_{0}^{t}\{\int_{0}^{s}g(\tau,y(\tau))d\tau\}ds=\int_{0}^{t}\{\int_{\tau}^{t}ds\}g(\tau,y(\tau))d\tau[/tex]

???????????????????? How do i prove that? should i do a change of variable? use fubini?

Im lost, pls help.

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# Q regarding Integration

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