$$f\left( x \right) = {x^n} \Rightarrow f\left( 1 \right) = 1$$

$$f'\left( x \right) = n{x^{n - 1}} \Rightarrow f'\left( 1 \right) = n$$

$$f''\left( x \right) = n\left( {n - 1} \right){x^{n - 2}}$$

$$ \Rightarrow f''\left( 1 \right) = n\left( {n - 1} \right)$$

$$\therefore$$ $${f^n}\left( x \right) = n!$$

$$ \Rightarrow {f^n}\left( 1 \right) = n!$$

$$ = 1 - {n \over {1!}} + {{n\left( {n - 1} \right)} \over {2!}}{{n\left( {n - 1} \right)\left( {n - 2} \right)} \over {3!}}$$

$$\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, + .... + {\left( { - 1} \right)^n}{{n!} \over {n!}}$$

$$ = {}^n\,{C_0} - {}^n\,{C_1} + {}^n\,{C_2} - {}^n\,{C_3}$$

$$\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, + ...... + {\left( { - 1} \right)^n}\,{}^n{C_n} = 0$$

$$F\left( t \right) = \int\limits_0^t {f\left( {t - y} \right)g\left( y \right)} dy$$

$$ = \int\limits_0^t {{e^{t - y}}} ydy = {e^t}\int\limits_0^t {{e^{ - y}}} \,ydy$$

$$ = {e^t}\left[ { - y{e^{ - y}} - {e^{ - y}}} \right]_0^t$$

$$ = - {e^t}\left[ {y{e^{ - y}} + {e^{ - y}}} \right]_0^t$$

$$ = - {e^t}\left[ {t\,{e^{ - t}} + {e^{ - t}} - 0 - 1} \right]$$

$$ = - {e^t}\left[ {{{t + 1 - {e^t}} \over {{e^t}}}} \right]$$

$$ = {e^t} - \left( {1 + t} \right)$$

$$y = {\left( {x + \sqrt {1 + {x^2}} } \right)^n}$$

$${{dy} \over {dx}} = n{\left( {x + \sqrt {1 + {x^2}} } \right)^{n - 1}}$$

$$\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\left( {1 + {1 \over 2}{{\left( {1 + {x^2}} \right)}^{ - 1/2}}.2x} \right);$$

$${{dy} \over {dx}} = n{\left( {x + \sqrt {1 + {x^2}} } \right)^{n - 1}}$$

$$\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{{\left( {\sqrt {1 + {x^2}} + x} \right)} \over {\sqrt {1 + {x^2}} }}$$

$$ = {{n{{\left( {\sqrt {1 + {x^2}} + x} \right)}^n}} \over {\sqrt {1 + {x^2}} }}$$

or $$\sqrt {1 + {x^2}} {{dy} \over {dx}} = ny$$

or $$\sqrt {1 + {x^2}} {y_1} = ny$$

$$\left( {{y_1} = {{dy} \over {dx}}} \right)$$

Squaring, $$\left( {1 + {x^2}} \right){y_1}^2 = {n^2}{y^2}$$

Differentiating, $$\left( {1 + {x^2}} \right)2{y_1}{y_2} + {y_1}^2.2x$$

$$ = {n^2}.2y{y_1}$$

or $$\left( {1 + {x^2}} \right){y_2} + x{y_1} = {n^2}y$$