Mother of All Calculus!

Calculus Level 4

If a positive real valued continuously differentiable functions \(f\) on the real line such that for all \(x\)

\[\large\ f^{ 2 }\left( x \right) = \int _{ 0 }^{ x }{ \left( { \left( f\left( t \right) \right) }^{ 2 } + { \left( f'\left( t \right) \right) }^{ 2 } \right) dt } + { e }^{ 2 }\]

is satisfied.

Then find the value of \(\large\ f(-1)\).

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