Download Solutions to Laplace Transform Problems: Pulse, Oscillation, and Shift Functions and more Lab Reports Differential Equations in PDF only on Docsity! Solutions to the Laplace Transform Worksheet. 1) > f_pulse:=3*(Heaviside(t-2) - Heaviside(t-4)); := f_pulse −3 ( )Heaviside −t 2 3 ( )Heaviside −t 4 > plot(f_pulse,t=0..5); 0 0.5 1 1.5 2 2.5 3 1 2 3 4 5 t > f_osc:=-sin(3*(t-1))*(Heaviside(t-1) - Heaviside(t-1-5/3*Pi)); := f_osc − ( )sin −3 t 3 −( )Heaviside −t 1 Heaviside − −t 1 5 π 3 > plot(f_osc,t=0..8); –1 –0.5 0 0.5 1 1 2 3 4 5 6 7 8 t > f_shift:=t*(Heaviside(t) - Heaviside(t-2)) + 2*Heaviside(t-2) + sin(3*(t-2))*(Heaviside(t-2) - Heaviside(t-2-2*Pi)); f_shift t ( )−( )Heaviside t ( )Heaviside −t 2 2 ( )Heaviside −t 2+ := ( )sin −3 t 6 ( )−( )Heaviside −t 2 ( )Heaviside − −t 2 2 π+ > plot(f_shift,t=0..10); > solution:=invlaplace(t3,s,t); := solution + + +( )−3 3 ( )cos −t 2 ( )Heaviside −t 2 ( )− +3 3 ( )cos −t 4 ( )Heaviside −t 4 ( )cos t ( )sin t > plot([solution,diff(solution,t), diff(solution,t,t)],t=0..5,color=[red,blue,black],discont=true); –4 –2 0 2 4 1 2 3 4 5 t 2) b) > de:=diff(y(t),t,t) + y(t) = f_osc; := de =+ d d2 t2 ( )y t ( )y t − ( )sin −3 t 3 −( )Heaviside −t 1 Heaviside − −t 1 5 π 3 > t1:=laplace(de,t,s); := t1 =− +s ( )−s ( )laplace , ,( )y t t s ( )y 0 ( )( )D y 0 ( )laplace , ,( )y t t s − − 1 3 e ( )−s + s2 9 1 1 3 e −s +1 5 π 3 + s2 9 1 > t2:=subs(sub,t1); := t2 =− +s ( )−s ( )Y s 1 1 ( )Y s − − 1 3 e ( )−s + s2 9 1 1 3 e −s +1 5 π 3 + s2 9 1 > t3:=solve(t2,Y(s)); := t3 + + + − −s3 9 s s2 9 3 e ( )−s 3 e − s ( )+3 5 π 3 + +s4 10 s2 9 > solution:=invlaplace(t3,s,t); solution − 1 8 ( )sin −3 t 3 3 8 ( )sin −t 1 ( )Heaviside −t 1 := − − 1 8 ( )sin −3 t 3 3 8 sin − +t 1 π 3 Heaviside − −t 1 5 π 3 ( )cos t ( )sin t+ + + > plot([solution,diff(solution,t), diff(solution,t,t)],t=0..10,color=[red,blue,black],discont=true); –2 –1 0 1 2 2 4 6 8 10 t 2) c) > de:=diff(y(t),t,t) + y(t) = f_shift; de + d d2 t2 ( )y t ( )y t t ( )−( )Heaviside t ( )Heaviside −t 2 2 ( )Heaviside −t 2+= := ( )sin −3 t 6 ( )−( )Heaviside −t 2 ( )Heaviside − −t 2 2 π+ > t1:=laplace(de,t,s); t1 := =− +s ( )−s ( )laplace , ,( )y t t s ( )y 0 ( )( )D y 0 ( )laplace , ,( )y t t s − + − 1 s2 e ( )−2 s s2 1 3 e ( )−2 s + s2 9 1 1 3 e ( )−s ( )+2 2 π + s2 9 1 > t2:=subs(sub,t1); := t2 =− +s ( )−s ( )Y s 1 1 ( )Y s − + − 1 s2 e ( )−2 s s2 1 3 e ( )−2 s + s2 9 1 1 3 e ( )−s ( )+2 2 π + s2 9 1 > t3:=solve(t2,Y(s)); := t3 − − − − − − − + +s5 9 s3 s4 10 s2 9 2 e ( )−2 s s2 9 e ( )−2 s 3 e ( )−2 s ( )+1 π s2 s2 ( )+ +s4 10 s2 9 > solution:=invlaplace(t3,s,t); solution − 1 8 ( )sin −3 t 6 3 8 ( )sin −t 2 ( )Heaviside − −t 2 2 π := − + − + 1 8 ( )sin −3 t 6 11 8 ( )sin −t 2 t 2 ( )Heaviside −t 2 ( )cos t t+ + + > plot([solution,diff(solution,t), diff(solution,t,t)],t=0..20,color=[red,blue,black],discont=true);
