Let $$f(x)$$ be a polynomial satisfying $$f(0)=2$$,f′(0)=3$$,f′′$$(x)=f(x) then $$f2(4)=$$

Question

Let $$f(x)$$ be a polynomial satisfying $$f(0)=2$$,f′(0)=3$$,f′′$$(x)=f(x) then $$f2(4)=$$

Infinity Learn · Accepted Answer

Given f′′(x)=f(x)2f$$′$$(x)f$$′′$$(x)=2f(x)f$$′$$(x)ddxf$$′$$(x)2=ddx{f(x)}2f$$′$$(x)2={f(x)}2+C$$ Now $$f(0)=2$$ and $$f(0)=3$$ …$$1$$ Equation from $$(1) f′(0)2={f(0)}2+C9=4+C$$⇒$$C=5$$∴f′$$(x)={f(x)}2+5$$ Now ∫$$1(5)2+{f(x)}2d{f(x)}=$$∫$$1dxlog$$⁡$$f(x)+5+{f(x)}2=x+C1f(0)=2$$⇒log⁡$$5=C1$$∴log⁡$$f(x)+5+{f(x)}2=x+log$$⁡$$5$$⇒log⁡$$f(x)+5+f2(x)5=x$$⇒$$f(x)+5+f2(x)=5ex$$⇒$$5+f2(x)+f(x)=5ex$$⇒$$5+f2(x)+f(x)=5ex$$ and ⇒$$5+f2(x)−$$f(x)=5e$$−x⇒$$f(x)=52ex$$−e−x⇒$$f(4)=52e4$$−e−$$4=52e8$$−$$1e4$$

$Let f (x) be a polynomial satisfying f (0) = 2, f^{'} (0) = 3, f^{''} (x) = f (x) then f^{2} (4) =$

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Let f(x) be a polynomial satisfying f(0)=2,f′(0)=3,f′′(x)=f(x) then f2(4)=

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$Let f (x) be a polynomial satisfying f (0) = 2, f^{'} (0) = 3, f^{''} (x) = f (x) then f^{2} (4) =$