Question

Solve the following second order differential equation analytically for x(t): - dx + 5x = 8 * 2 for the following two cases: Case 1: all initial conditions are zero. Case 2: given the initial conditions: x (0) = 1 (0) = 2 For both cases, also plot the solution obtained, for t = 0 to 10.

please write the code for the plot

Answer 1

d x +7 ds +7 dx + 5x = 8. d t - dt It is in form of second order differental equation General solution is the sum of and complementry solution Partical solution . The complementry solution is given by dag - + 7 dx + 5x = 0 - 0 dt² at det oc(t) = eft de for dett, a x-xett d + 2 substute dx 1. dx : dt in eqo ☺ dt

2 et +7 de At+sento ett (d² + 7 +5)=0 solve ² + 71 +5=0 =) = 72 + 4 General sowlion for Quadratic equation is x= -btJ6²-4ac 2a . de -7+49-4((5) 211) d = - / + 0 29 , d = 2/2 - 1 29 .:.2017) - Ge 12-)+, x2605 67 671127) + Where C1, C2 are arbitrary constants General solution is gi(t) + X(t) = X(t) .: X(t) = Ge (7-7)4 64 47 12T) +

The pirticular solution, a is constant & xp (t) = a, dap (t) = 0, dt è p(t) = 0 - d t .: the equation is 0 + 700) + 50 = 8 sa =8 ar = 8/5 .: the requsied solution is 10 [459] GETTI

x= 6, 66:7-). Get egye (2 LET)a ele29). CCE MET DE Sitial conditions, 2 (0) = 0, x'(o) = 0 at x (0)=0 x'(o)=0 0= 61+ (7 +8,162 -V27) C: +42 +129) 4 =0 on solving citius (-29+7 379) 145 Gas (29 +7527) 145 .: requsied solution, (129)+ 2= (-7+79) + (29+7.57a) e + 8 ius (29+7/29 = 6297729) - 45

( 7 ، پی ( تکی، و ) ب( ام و ( تاء (واده ام ( ع اه conditions, x (0)=1, sc (0)=2 Inital 2 - (acco , ا= (c co< لمه ه- و ( 2 ) + ( 2 ) + + G =1 on solving, (4529 87) و : )| 27 -021) C1 = 50 solution, .: requsied و کہ ( اے (1420) = 7 200 200