Question

Find a particular solution to the differential equation using the Method of Undetermined Coefficients. dy dy -5 + 2y = x e* dx? dx A solution is Yp(x) =

Answer 1

d²y dx2 - 5 dy SOIN! + 2y = xex particular solution to the differential equation y"- 5y' +2y = 0 el? - 591 + 2 = 0 25-8 34= 5 + 525-4x2 2 4= 5+ 2 es=51517 2 et 4L 15+517 Yn = cel 5-517) + Cad 9) fund yp Yp - cea ARAB Bot ce? (AX+B) cenc Ax+B) - @? (AC X + BC) Vp (t) = e( Ax+B), YP(x)= a[x et +eXJ +BeX = Axe? + (A+B) ex ex YpCX)= A[zeltex] +(4+B)ee Yp'(x)= Axe? +(2 A+B)e? Axe? +42 A+B)e?- 5 [[Axe? + Aet) +B0?] + 2 [re? A + Belj = 10% Axe? 5 AXET+2xé Acer (A+B) e*-5(A+B)ex': ex QA+B-5 A-5B +2 B = 0 +2B ex -3A-2B = 0 -4A +2 A= 1 -2A=1 TA = 1/2 B = 3 4 ep-2 +3 - 3*-|-2B 3x12 3=B 2X2 y p(x) = 4

d²y dx2 - 5 dy SOIN! + 2y = xex particular solution to the differential equation y"- 5y' +2y = 0 el? - 591 + 2 = 0 25-8 34= 5 + 525-4x2 2 4= 5+ 2 es=51517 2 et 4L 15+517 Yn = cel 5-517) + Cad 9) fund yp Yp - cea ARAB Bot ce? (AX+B) cenc Ax+B) - @? (AC X + BC) Vp (t) = e( Ax+B), YP(x)= a[x et +eXJ +BeX = Axe? + (A+B) ex ex YpCX)= A[zeltex] +(4+B)ee Yp'(x)= Axe? +(2 A+B)e? Axe? +42 A+B)e?- 5 [[Axe? + Aet) +B0?] + 2 [re? A + Belj = 10% Axe? 5 AXET+2xé Acer (A+B) e*-5(A+B)ex': ex QA+B-5 A-5B +2 B = 0 +2B ex -3A-2B = 0 -4A +2 A= 1 -2A=1 TA = 1/2 B = 3 4 ep-2 +3 - 3*-|-2B 3x12 3=B 2X2 y p(x) = 4