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Ch 6 theorems for exam 2

{\bf Section 6.1}

{\it Thm  6.1.2: }

Hint:  Show $\int_0^\infty e^{st}f(t)dt$ exists by showing 
$lim_{A \rightarrow \infty} \int_0^A e^{st}f(t)dt$ exists (ie converges to a finite number) for $s > a$.

Note $ \int_0^\infty e^{st}f(t)dt =  \int_0^M e^{st}f(t)dt +  \int_M^\infty e^{st}f(t)dt$

{\it Thm:}  The Laplace transform is a linear operator.

Hint:  ${\cal   L}(af(t) + bg(t)) = ...$  OR 
[ ${\cal   L}(af(t)) = ...$  and
 ${\cal   L}(f(t) + g(t)) = ...$]


{\bf Section 6.2}

{\it Thm 6.2.1:  }

Hint:  use integration by parts and let $dv = f'(t)$

{\it Cor 6.2.2': } ${\cal L}(f''(t)) = s^2{\cal L}(f(t)) - s f(0) - f'(0)$

Hint.  Use that $f''$ is the derivative of $f'$.  Let $g = f'$ and note  ${\cal L}(g'(t)) = s{\cal L}(g(t)) -  g(0) $

{\it Table 6.2.1}

{\bf Section 6.3}

{\it Thm 6.3.1}

Hint:  $ \int_0^\infty  h(t)dt =  \int_0^c h(t)dt +  \int_c^\infty h(t)dt$ and use $u$-substitution (let $u = t-c$).

{\it Thm 6.3.2}

Hint: Let $F(s) = {\cal L}(f(t))$. Use definition of Laplace transform to evaluate 
 ${\cal L}(e^{ct} f(t))$ and $F(s-c)$.


Better 
Hint: Let $F(s) = {\cal L}(f(t)) = ......$.  To calculate $F(s-c)$ 
evaluate $F(s)$ at $s-c$  (i.e. replace $s$ with $s-c$).
 Use definition of Laplace transform 
to evaluate 
 ${\cal L}(e^{ct} f(t))$.

{\bf NOT on exam:}

Thm:  If $f$ is a bijective linear function, then $f^{-1}$ is also a linear function.

Cor:  ${\cal L}^{-1}$ is linear.


Cor 6.2.2 

Hint:  Use proof by induction.



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