# 352 - Topic 14 - Laplace Transforms

## 3299 days ago by Professor352

Laplace Transforms and Sage

John Travis

Mississippi College

%auto t,s=var('t,s')
# Find the laplace transform of f(t)=1 f(t)=1 assume(s>0) F(s)=integrate(e^(-s*t)*f(t),t,0,oo) html('$F(s) = %s$'%str(latex(F(s)))) (plot(f,(t,0,2),color='blue')+plot(F,(s,0,2),ymax=20,color='green')).show()
 $F(s) = \frac{1}{s}$
# Find the laplace transform of f(t)=t f(t)=t assume(s>0) F(s)=integrate(e^(-s*t)*f(t),t,0,oo) html('$F(s) = %s$'%str(latex(F(s)))) (plot(f,(t,0,2),color='blue')+plot(F,(s,0,2),ymax=20,color='green')).show()
 $F(s) = \frac{1}{s^{2}}$
# Find the laplace transform of f(t)=t^n assume(s>0) L2 = integrate(e^(-s*t)*t^2,t,0,oo) L3 = integrate(e^(-s*t)*t^3,t,0,oo) L4 = integrate(e^(-s*t)*t^4,t,0,oo) L5 = integrate(e^(-s*t)*t^5,t,0,oo) L6 = integrate(e^(-s*t)*t^6,t,0,oo) L7 = integrate(e^(-s*t)*t^7,t,0,oo) G = Graphics() assume(s>0) for n in range(2,8): f(t)=t^n F(s)=integrate(e^(-s*t)*f(t),t,0,oo) G += plot(f,(t,0,2.5),color='blue')+plot(F,(s,0,3),ymax=20,color='green') html.table([[t^2,t^3,t^4,t^5,t^6,t^7,' '],[L2,L3,L4,L5,L6,L7,' '],[' ',' ',' ',' ',' ',' ',G]])

 $t^{2}$ $t^{3}$ $t^{4}$ $t^{5}$ $t^{6}$ $t^{7}$ $\frac{2}{s^{3}}$ $\frac{6}{s^{4}}$ $\frac{24}{s^{5}}$ $\frac{120}{s^{6}}$ $\frac{720}{s^{7}}$ $\frac{5040}{s^{8}}$

var('n') assume(n,'integer') assume(n>1) laplace(t^n,t,s).show()
var('b') S = sin(b*t) C = cos(b*t) LS = laplace(S,t,s) LC = laplace(C,t,s) f(t)=sin(t) assume(s>0) F(s)=integrate(e^(-s*t)*f(t),t,0,oo) G = plot(f,(t,0,2*pi),color='blue')+plot(F,(s,0,2*pi),ymax=2,color='green') f(t)=cos(t) assume(s>0) F(s)=integrate(e^(-s*t)*f(t),t,0,oo) H = plot(f,(t,0,2*pi),color='blue')+plot(F,(s,0,2*pi),ymax=2,color='green') html.table([[S,C],[LS,LC],[G,H]])

 $\sin\left(b t\right)$ $\cos\left(b t\right)$ $\frac{b}{b^{2} + s^{2}}$ $\frac{s}{b^{2} + s^{2}}$