Question

solve using laplace please.

xy'' + (1-x)y' + 2y = 0 , y(0) = 1 , y'(0) = -2

Answer 1

Given equation is. xy"+(1-2) Y'tayo yeo) = 1, y'60) = -2. 3 -5 SCS-1) ay" + y - xy + 2y = 6 we know that Llyn = s²yes) - syo) - y'(0) er from the property of laplace transform, an it L[y] = Y(S) Out L[xy] - des yes) - Apply caplace trams form to above eauation on both sides, [xgv] + LLY!]- 2(xy!] +aL[y]=0 => L [ Y"] + [CY] - (204] + a2 CY] = 0 --[sayo) - syo) - yoi] + Sycs)-40) td [syes) - Yo] + 2 4 Ls) = 0 - d szycs) - +2] + Sy(s) -1 + [sycs) -17 +Q YOS) =0. 3-5 O ds s²y's) gy'CS) & 25 YCS) -(-1) -O + SYIS) 1 S + sy'cs) & YCS) (11-0 + 2y (6)=0.

- y'CS) [-s²+s] + y(s) [-s +3] +1-1 so. yles) (-5°+s] + y) (-s+] = 0 yyles) ( S2-s]+ yes) [S-3) = 0. => Y [SP-] - Y(S) [3-5). dy 3-5 de (?k_s) 3-5 S(5-1) A - 3-5 3-5 A-t B SCS-1) s sel 3-5 = ACS-1) + BS ( s for s=0 i 0 3= -A - A=1- 3 for s=1 2= B B=2. Integrate both sides, Inly) = -3 ens ta ln (5-1) + Inc. where Inc is constant.

- lny = ln s 3 + ln (S-1)2 + lnc any en (c(s.1²) .: yes) - co Yes) = ( (5²_25+1) 53 yes) =[s B transform on both sides laplace Inverse By applying yox) = [1-24 + 8) y(x) = c [ 1 - 2x + x2] Given y 10) = 1 yeo) = c[i] + c = 1 900) = 1- 21+ 3x2