Question

Answer ALL the questions. Some or all of them shall be marked. Question 1. Consider the following system of differential equations: P.(D) [x] + P (D)) -(0) Px(D) [x] + P (D) x = f(t). (1) How do we determine the correct number of arbitrary constants in a general solution of the above system. (0) Explain briefly the difference between the operator method and the method of triangu- larization when used for solving the above system. Question 2. Determine whether the follwoing systems of differential equations are degener- ate. In the degenerate cas, decide whether has no solution or infinitely solution. If it has no solution, explain why, else solve y(t) in term of (t), where r(t) is an arbitrary function of t. (a) (Exercise 1.26 (2) of SG) i-+-1 (b) -ty-y-3e- + 2x + y + 2y = 3 Question 3. Use the operator method (method of elimination) to find a solution of the follow- ing system of differential equations: NB Eliminater first 15 Question 4. Reduce the system of equations: is equivalent to the following triangular systems: PD) -(0) B(D+B(D) -(0) and Determine the Solution Question 5. ( Suppose X is an eigen value of A of multiplicity k > 1. Define a root vector corresponding to the eigen value .. (6) Use the induction principle to prove that if eMU,,' U'U, are solution corre- sponding to then distinct eigenvalues 11, 12,..,, of an n x n matrix A, then ,' U'U, are linearly independent. Question 6. Use the eigenvalue-sigervector method to solve the initial value problem 21 -1 X-0-1 0 0 X X(0) - 20 --10 15 Question 7. Find a general solution of 2 1 6 X = 0 25x 0 0 2 -C: by using root vectors. Question 8. Read Theorems 2.18,3.8, 3.15 and Lemma 3.10, 3.11 carefully and their Proves.

Answer 1

1)

i) Let P. (D) PilD) Pan (D) The number of arbitary constant is equal to the degree of s. ID Pn (D) [a] + Plz (D) [y] = f(t) P21 (D) [ay & P22 (D) [y] falts when operator method is ured either y is immediately eliminated by either uring the operations. P21 (D) [1] - PuCD) [2] to eliminate a Pan (D) [1] Pia (D) [2] to eliminate y In method of triangularization, we. consider all polynomials. PCD) and retain. the equation in which the order of PCD) is the lowest Instead of eliminating a or y all together in one step redune in each step the order of the coefficient polynomial Pu CD) of a (for example) by one until the coefficient polynomial of s constant. This equation is then wred to find an expression of a in terms of y which is then substituted into the and equation to find an equation only in iny leads to a triangular system which a consist of one equation in in y and another where the order of the has been coefficient polynomial of . reduced to a contant. we and y

2)

a)

x'-x+y'=t       -----eq1

x''-x'+y''=1    ----eq2

eq 2 is differentiaon of eq1

these can be written as

(D-1)x+Dy=t

(D-1)x'+Dy'=1

operational determinant is:

(D-1)D-(D-1)D=0

so the given system is degenerate .

and has infinite solution.

b)

solution - Given sistem ich differential equations x-x+ g-y=-30 2t -2t .....i and x + 2x tý + 2y = zet first of all, the equations (i) and (ii) can be written in symbolic form as (0-1) x + (0-1)y=-30 ...(ii) and (P+2)x+ (8 +2y = 3 et ...(iv) where Now for degenerated solution I fo(D) fa (D) 3,(7) 7(1) Here from equation (ii) and (iv) fo(a) = D1 and g, (D)= 8+2 fz(D) = -1 J.(P) = 8+2 D7 DI therefore =0 +2 D+2 Hence the given system of equations are degeneratef. Equation iii) and (iv) have infinitly many solutions.

NOW and Pluty) = -(2x+2y+zet - (vi) solve the equation and (iv) vir - solidslas (P-1) 4+ (9-12y = .-zert and (+2) 4 + (1+2)) = 34t Above equation written as - Equate equition crj and (vi) platy) = x+y-Bert - (v) 3(ety) = sett att -2t Inty = etté Hence the given system of differential systems is degenerated and has infinite many solutions satistying Ars x+y = ettet

3)

- 4*- Guien diffetential et au (terat) ie do 40[ "too Y. and do -ada d (+9+ D2 = Yo- 9 Ho (1+00)" +72+) wb [-90+ -] [ 2°+2+] [(tºhat) n ( 3 ) to [(#21) - 11 )] [ ++22-2 - 4 1 1 2 t - 2) dt de Use D forse & Becomes Horte Bytt > Dx-427d'y = - Dy + (0-4) 2+ and by : - -2-Dx Dy + Dx+x=0 Dy+ (D+1)x=0 ☺ To Eliminate 35, Mult, By (D+) and Ⓡl8y CD-u) & subtracting dt D%(D+)y + CD-u) (D+3}<= (D+1)** D (D-4y (D-Y(D+1) X = 0 [p*CDH) – D (D-WJy = (0+) t3 + t² t 19 u 8 + 10 + 0² - 8² +4D) y = Dt + t² (ft P.g. (0² + 4D) y = 2t+t 9ts A.ES D + 4D=0 C, 4 (2 Cosat +(3 Sinat t D(0²+4)=0 D=0, 0²-4 + E t 4 8 D=0, D=+ di cf = Geot + C2 Cosat + Zsinat rahat is the required Pig, Son D3 +4 D 2 O 2 외스 3 12.

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