Question

Please help me out with question 18.

Its a question of laplace transfromation for unit step functions .

thanks

lems. Sketch a graph 14. F(s) = e-s (3) +e-> (5) Use Laplace transforms to solve the following initial value problems. Sketo the solution, together with a graph of the driving function. 15. x' + 2x = 2u(t - 3), x(0) = 1 16. x' + x = 2 + ut - 1)e-!, x(0) = 0 17. x" + 4x' + 4x = u(t - 2), x(0) = 0, x'(0) = 1 18. x" + 2x' + 5x = 2 + uſt - 2), x(0) = 0, x'O) = 0 vidar an RC-circuit with input voltage given by the fr might have wh

Answer 1

"'+ex' +5X = 2 + ult-2), X10) = 0, x'(0) = 0..-11) (9) = y(s), f/Y') = syis) - 910), 8 (99) = sº y15)-576)-y(0) I (461)) = e.es transform both sides of equation (2) taking laplace we get L(x")+22(x) +58(x) = 2(2) + 2 144-2)) s'X19) - SX10) - Y10) + & SX(s) - 2X10) + 5X75) e-25 5°x45) +25415)+5465) = a + emas (52+25+5)x(s) = { rete 200) - vétség) ****13:45) By partial traction method we get s(5²+25+5) 5/5²+25+5) sta (5+1 ²+ 4

(5+2) 5(5²25+5) s(s²t2 S45 stiti 5 (5²+25+5) sti X (5) = 2013 febrer (asyon) *cept te-25 taking inverse laplace transform both sides we get 2.0) (cte*it totime) aug 4 1 e 2-t coopt-4) - į e² tsine *4 =} ('-etcost-e triol) +1.1)/1-22 to ou e) - > 5T ce-tgio tot-2). x64) - (-e-tecost-e-triort) + 0.117/1-e2 tos (ott) - center