Question

III) Consider the following differential equation: 5i(t) + 3r(t) - 4 = 0, 7(0) = 2. 1. Find the backward solution. 1,5 mark 2. Is this solution convergent or divergent? Justify your answer. mark 3. Determine the stationary solution and indicate whether it is stable or unstable. 1 mark 4. Sketch a phase diagram and a time-path diagram. 1,5 mark

Answer 1

Given equation t. Sült) + 3 xlt) -4 = 0 Let d dt = D, then えした) Drelt) In operator form, our equation becomes SD ult) + 3x1) 4 (SD +3) xlt) 4 Auxiliary equation es. 52 +3 -0 the complementary function ts. rec cett rec set Particular solution. 4 Xp 10 et 5D +3 ap م (م)

Hence the general solution es. relt) = ne tep - 3/5 t x (+) = cie + wie Now using nco) = 2 2 c, e + 4 3 ci = 2- ㅠ wif 235 -31st relt) - WIN e + w/f 2 lim 3/5 lim t-00 alt) ( +) tyoo 1 now - 2 Lim 3 tto 2/3 Lb e + ( ما = N(م) + XO wif lim t 700 nit) 12 wlf finite value so, the solution es convergent.

(3 the equation ③ Rewriting silt) = 3 7 Lt) + 5 RHS must for the stationary solution, be equal to zero + mih ult TT Wif 2- a stationary solution. let fin) = - 3 re (t) + 4/5 1 f'() = y/W al- es. - } <o t'(a) at wlf ala : f') 20 20, 80 A(ن) is stable Phase diagram x = 42

time-path diagram. 2 2 . = 1.33