MATH 202 Exam 2 with Solution

MATHEMATICS 202 ( lst Semester, 1994-95 ) Quiz I Prof. N. Haidar FACULTY OF ARTS & SCIENCES, AUB Time : 60 mins. January 18,1995 (closed book) I. Solve the following DE's. (a) x? y''—4xy'+6y = Inx? @) xy"+(1-x)y'=0 I. Use the substitution ¥ = vx u(X) to solve x7 y"+(x? +1/4)y=0 J -20 III. Represent 2° in terms of elementary functions . IV. Find the Laplace transform of f(t) and of f(t) U(t-2) when { 0, o<t<1 f= t, t21 1) u,= Solution QuizII 1994-95 2. x’ y"—4xy'+6y = Inx? 2,0 x y"—~4xy'+6y = 0 0 singular point r(r-1) - 4r+6=0 r2-S5r+6=0 => r=] & r=6 but we can easily see that this is a caushy-euler equation => y=ox+c,x* Yp=?? w= 5x° w, =—2x° Inx w,=2xInx w)= “ Ww -2 —Inx => u, =2xInx-2x w= ane 5 Ww, 3 2 Yip = 5 (xInx—x)x Yap b) = : Jo? In x)dx by parts we get : xy"+(1-x)y'=0 Letty =z=>y"=2’ xz’ + (1-x)z=0 (1-x) x . r(r-l)+r=0 = r=0 Z= ¥C,x™", 2'= S(tyc,x ex: | xT (nt r)C,x"-C,x°+ Cx = Vi(k+nC,x* -C, jx + VC, x* k=0 k=I ‘ tC, =0 C=C, P= °o x=1-x => Oisa singular point 2) 3) 4) Oe substituting in (1) & c,= a1, 0C,=0=C k-1 z=0 > Yi -C, ~ {Max 1 e° * 1 ;e* =C dx = dx Y2 oJ C,” x Cy J 2 y=vx u(x) x? a +1/4)y=0 (1) Yeager (x)Vx wt ue?) + w" Wx + x7u''(x) + xu' (x) + x?u(x) = 0 J 20 2 2J, =xJ 44 + x3.) 2 —xJ . J 42 Ee V2 J,= = sinx a WG) multiplying by x-!/2 we get : which | is a Besselle equation where v = 0, y = C,J,x + CYox