Question

1 III) Consider the following differential equation: 5:(t)+32(t) - 4 = 0, 2(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

22 Given couation is sit)+3xt)#420-> Let D = d d then silt= da ort) In the operator form, our equation change aus SDX) +38(t)=4. → SD+3) r(t)=4 Auxilary equation is Sa+30 >> sx=-3 t=- complementary function is de=cent -31st &c=ge particular solution is - 니 50+3 eat ap=43 Therefore the general solution is at) - actre ant) = -31st се. ty Now by using no=2 2=.48 ty G=2-4 = 2 / +4 3 [i:26) 35e-/st (ori --3/5t st to go xt)= at to (fel + GIM 2 Lim tto to المر عاما = 2 x 0 + 4 / 4 (finite value Lim X(t) = 43 هم دt Hence, the solution is convergent

3 Now by rewriting the equation si(t) 기봉 > for obtaining stationary solution, RHS need to be equal to zero 문라 at 20 크문가 봉 5 - M 스볼 3 * = 4 is a stationary solution let f(x) = 중기빠는 flas +6 f'(x) f (4) at nu is 320 Hence x=y is stable : () o

19 Now let us draw the phase diagram, first xay 3 > Time-path Diagram or on tone of x=1-33 1 t