Math 15 note Laplace transform

 ANNIHILATOR APPROACH

Given {\displaystyle y”-4y’+5y=\sin(kx)}, {\displaystyle P(D)=D^{2}-4D+5}. The simplest annihilator of {\displaystyle \sin(kx)} is {\displaystyle A(D)=D^{2}+k^{2}}. The zeros of {\displaystyle A(z)P(z)} are {\displaystyle \{2+i,2-i,ik,-ik\}}, so the solution basis of {\displaystyle A(D)P(D)} is {\displaystyle \{y_{1},y_{2},y_{3},y_{4}\}=\{e^{(2+i)x},e^{(2-i)x},e^{ikx},e^{-ikx}\}.}

Setting {\displaystyle y=c_{1}y_{1}+c_{2}y_{2}+c_{3}y_{3}+c_{4}y_{4}} we find

{\displaystyle {\begin{aligned}\sin(kx)&=P(D)y\\[8pt]&=P(D)(c_{1}y_{1}+c_{2}y_{2}+c_{3}y_{3}+c_{4}y_{4})\\[8pt]&=c_{1}P(D)y_{1}+c_{2}P(D)y_{2}+c_{3}P(D)y_{3}+c_{4}P(D)y_{4}\\[8pt]&=0+0+c_{3}(-k^{2}-4ik+5)y_{3}+c_{4}(-k^{2}+4ik+5)y_{4}\\[8pt]&=c_{3}(-k^{2}-4ik+5)(\cos(kx)+i\sin(kx))+c_{4}(-k^{2}+4ik+5)(\cos(kx)-i\sin(kx))\end{aligned}}}

giving the system

{\displaystyle i=(k^{2}+4ik-5)c_{3}+(-k^{2}+4ik+5)c_{4}}

{\displaystyle 0=(k^{2}+4ik-5)c_{3}+(k^{2}-4ik-5)c_{4}}

which has solutions

{\displaystyle c_{3}={\frac {i}{2(k^{2}+4ik-5)}}}, {\displaystyle c_{4}={\frac {i}{2(-k^{2}+4ik+5)}}}

giving the solution set

{\displaystyle {\begin{aligned}y&=c_{1}y_{1}+c_{2}y_{2}+{\frac {i}{2(k^{2}+4ik-5)}}y_{3}+{\frac {i}{2(-k^{2}+4ik+5)}}y_{4}\\[8pt]&=c_{1}y_{1}+c_{2}y_{2}+{\frac {4k\cos(kx)-(k^{2}-5)\sin(kx)}{(k^{2}+4ik-5)(k^{2}-4ik-5)}}\\[8pt]&=c_{1}y_{1}+c_{2}y_{2}+{\frac {4k\cos(kx)+(5-k^{2})\sin(kx)}{k^{4}+6k^{2}+25}}.\end{aligned}}}

{\displaystyle y_{p}={\frac {4k\cos(kx)+(5-k^{2})\sin(kx)}{k^{4}+6k^{2}+25}}}  

{\displaystyle y_{c}=e^{2x}(c_{1}\cos x+c_{2}\sin x)}.

Laplace Transform

Sufficient Conditions for Existence
f(t) is piecewise continuous on the interval [O, oo) and of exponetial order

F(s) ->0 as s -> oo

		conditions
1

Properties of the unilateral Laplace transform

	Time domain	s domain	Comment
Linearity	{\displaystyle af(t)+bg(t)\ }	{\displaystyle aF(s)+bG(s)\ }	Can be proved using basic rules of integration.
Frequency-domain derivative	{\displaystyle tf(t)\ }	{\displaystyle -F'(s)\ }	F′ is the first derivative of F with respect to s.
Frequency-domain general derivative	{\displaystyle t^{n}f(t)\ }	{\displaystyle (-1)^{n}F^{(n)}(s)\ }	More general form, nth derivative of F(s).
Derivative	{\displaystyle f'(t)\ }	{\displaystyle sF(s)-f(0)\ }	f is assumed to be a differentiable function, and its derivative is assumed to be of exponential type. This can then be obtained by integration by parts
Second derivative	{\displaystyle f”(t)\ }	{\displaystyle s^{2}F(s)-sf(0)-f'(0)\ }	f is assumed twice differentiable and the second derivative to be of exponential type. Follows by applying the Differentiation property to f′(t).
General derivative	{\displaystyle f^{(n)}(t)\ }	{\displaystyle s^{n}F(s)-\sum _{k=1}^{n}s^{n-k}f^{(k-1)}(0)\ }	f is assumed to be n-times differentiable, with nth derivative of exponential type. Follows by mathematical induction.
Frequency-domain integration	{\displaystyle {\frac {1}{t}}f(t)\ }	{\displaystyle \int _{s}^{\infty }F(\sigma )\,d\sigma \ }	This is deduced using the nature of frequency differentiation and conditional convergence.
Time-domain integration	{\displaystyle \int _{0}^{t}f(\tau )\,d\tau =(u*f)(t)}	{\displaystyle {1 \over s}F(s)}	u(t) is the Heaviside step function and (u ∗ f)(t) is the convolution of u(t) and f(t).
Frequency shifting	{\displaystyle e^{at}f(t)\ }	{\displaystyle F(s-a)\ }
Time shifting	{\displaystyle f(t-a)u(t-a)\ }	{\displaystyle e^{-as}F(s)\ }	u(t) is the Heaviside step function
Time scaling	{\displaystyle f(at)}	{\displaystyle {\frac {1}{a}}F\left({s \over a}\right)}	{\displaystyle a>0\ }
Multiplication	{\displaystyle f(t)g(t)}	{\displaystyle {\frac {1}{2\pi i}}\lim _{T\to \infty }\int _{c-iT}^{c+iT}F(\sigma )G(s-\sigma )\,d\sigma \ }	The integration is done along the vertical line Re(σ) = c that lies entirely within the region of convergence of F.[15]
Convolution	{\displaystyle (f*g)(t)=\int _{0}^{t}f(\tau )g(t-\tau )\,d\tau }	{\displaystyle F(s)\cdot G(s)\ }
Complex conjugation	{\displaystyle f^{*}(t)}	{\displaystyle F^{*}(s^{*})}
Cross-correlation	{\displaystyle f(t)\star g(t)}	{\displaystyle F^{*}(-s^{*})\cdot G(s)}
Periodic function	{\displaystyle f(t)}	{\displaystyle {1 \over 1-e^{-Ts}}\int _{0}^{T}e^{-st}f(t)\,dt}	f(t) is a periodic function of period T so that f(t) = f(t + T), for all t ≥ 0. This is the result of the time shifting property and the geometric series.

{\displaystyle f(0^{+})=\lim _{s\to \infty }{sF(s)}.}

https://www.youtube.com/watch?v=OiNh2DswFt4&list=PL96AE8D9C68FEB902&index=26