MATH 202 Exam 2 with Solution

MATHEMATICS 202 ( 2nd Semester, 1992-93 ) Quiz I May 13, 1993 1) Apply the Frobenius method to solve the equation : Zoe x? y''-2xy'+(2 - x*)y =0 and identify the series obtained as expansions of known functions . 2) — Solve the initia] value problem : x'=-2x+y y' =-4x + 3y + 10 cost with x(0)=1; y(0)=-6 3) Use the Laplace transform method to solve : y'(t) - 4y'(t) + 4y(t) = 4 cos 2t with y(O)=2; y'(0)=5. 4) Consider the differential equation : y't+4x*y =0 .. (A) Use the substitution y(x) = x*u(x) where u is a function of x, and find the differential equation satisfied by u. Then determine a so that the equation satisfied by u has the form : x?ul'+xu'+(4x"° -1/4)u=0 cee (B) Make the substitution z= bx° and determine b so that equation (B) is turned into a Bessel equation (C) . c) Deduce from parts (a) and (b) the general solution of equation (A) . Solution QuizlT 1992-93 a2. ot 1) x’ y"-2xy'+(2 — x7)y =0 xp(x)=-2 ; x2q(x) =2- x2 atv=2 0 is a singular point . r(r-1) + por + qg = 0 2-3r+2=0 => r=l & r=2 yada” 5 y= (atr)cx""! n=0 n=0 y= Vat ry(ntr—VNeo,x? n=0 (S (ntry(ntr—Ne,x" -2>0 (atr)e,x” +2>.¢,x" - Sex?) ° n=0 n=0 n=0 for the firstthreen=k & n+2=k=> n=k-2 «(Se +ryktr—Me,x* -2>(k+r)e,x* + 2>.c,x! _ So,.2x*] k=0 k=0 k=0 k=2 r(r-1)- 2r+2=0 ; r=1,&r=2 r=2 => 2c) =0=>cl1=0 r=l1=> 0cl1=0 undetermined (k = 2) ((k+r)(k+1-1) - 2(kt+r) + 2]c, +CK.2 =0 C, = “1-2 ; for (r = 2) (k+r\(k+r—3)+2 c= Cy-2 oo (A +2)(k-1) +2 C,=¢,/6 ; ¢,=0 3; c,=c,/120 ; c,=0 7) Cy, = (2n+ 1)! y, =xsinx ge LPesae yr =y f 7 dx = ~xsinxcot gx y 2) x =-2xt+y y' =-4x + 3y + 10 cost y" =~4x' + 3y' -10sint y" = 8x - 4y - 12x +9y - 10sint £3 3) 4) y" =yi - Ty - 10(cost + sint) y"-y' + 7y=0 m2-m+7=0 A=- 1 V6. 1 v6. m=--—i; m= ti 2 2 i ee y =e (c, cost +C, cos) y, = Acost + Bsint solving and comparing with the origina] eqn. we get : A=5/37 ; B= -70/37 12 v6 V6 (c, cos “yf +e cos 1) + Acost + Bsint General Solution = ¢ y(t) - 4y'(t) + 4y(t) = 4 cos 2t yO)=2; yi(O)=5. Cy" - 4y' + 4y) = C(4cos2t) s’y-—2s—s—~ 4sy+ 8+4y=— 5 4s s +4 yet +( 4s 2873 (s* +4)(s? —4s+8) s*-45+8 . 3 . y = cos2t x 2sin2te”' +2e*' cos2t 5" sin2t Compulsive product. y''+4x*y =0 .. (A) y(x) = x%u(x) a ax u(x) + ul(x)x" "= a(a —1)x*? u(x) + +u'(x)(ax7") + u"(x)a + au'(x)x7" im x*u''+2au'x*! + a(a— I)x* ut 4x7*u = 0 let a= 1/2 xu x? + u(dx®? —(1/4)x) multiply by x*” x?ul+xu'tu(4x'? — 1/4) 6Y b) z=bx* az = 5bx’d x Ou @uax yu du ~ = 7 => uw = Sbx" — 0.z Qxdz 5bx Oz Oru _ (6 u'/Sbx')0 x _ u''(5bx*) — 20bx7u' ; 0°z Ox dz (Sbx*)? Sbx! 2 oH (sbx*)? x x? + (20bx2u') x x? + 5bx* + u(x)(4x!” ~ 1/4) 2 Zz 2 2H a5p2x" + 20bx* + Shx*) oY + u(xy(4x" - 1/4) Oz az letb = 5 > 2u!+zu'+u( 2-2? —1/4) : 25 c) Bessel equation 2 2 u= ain) +O,J ip (G?)