5. Solve the following using "Reduction of order." (20 points) y" ty = exsin (x) step 1 : Homogeneous solution + 9 = 0 re + E 2 ( 1 ) 2 r= ofi 2 + Bi d = 0 Cosx Sinx r=o-i B = 1 - sinx cosx y1 = cos ( X ) 1 yz = sin (x ) w (y , , 4 2 ) = 1fo yh = CI cos ( x) + cz sin ( x ) Step 2 : reduction of order y = usin (x ) "! = u'sin ( x ) + ucos ( x ) y " = u" sin ( x ) + u' cos ( x ) + u cos ( x ) - usin(x ) = u" sin ( x ) + zu'cos ( x ) - usin(x )gul + y = ex sin (x ) u" sin ( x ) + 2 n'cos ( x ) - USin ( x ) + usin (x ) = exsin(x ) u"'sin (x ) + 2u'cos( x ) = exsincx ) make w = u , w = u sin (x ) w + 2 605 ( x ) W = ersin (x ) w / + 2 Cos ( x ) W = Px sin (x ) P P = 2LS(X) 9 = ex Sin(x ) u = Sin (x ) 1 . F . = p ) pd x du = cos ( x ) . dx 2 WS (x ) e sin(x ) dx = e 2 \\ u = ( 2ln / sin(x )] = Sin (x ) ugd x sin(x ) . ex dx u = W = u Sin 2(x ) Computer = ex 2 excos (2x ) exsin ( 2x ) + + 5 Sin (x ) 5 sin? (x ) Sin'(x )u = Hex + 2 ex cos ( 2x ) exsin ( 2x ) + CI dx 5 Sin? ( x ) 3 sin 2 ( x ) Sin'(x ) = (ex + zex (1-2sin(x)) _ e" (2sin (x) (s(x ) + CI 5 sin ? ( x ) sin 2 (x ) )dx 5 sin ? ( x ) = et ex dx - 4 ex _ 2 [ ex cos(x ) dx CICOS( x ) sin ? (x ) 5 ) sin ( x ) Sin(x) = 5 ex - 2 ex cosix _ CICOS( X ) Computer 5 + C2 Sin(x ) Sin(x ) Step 3 : General solution : y = usin( x ) As a whole = ex 2 ex cosix ) CICOSLX ) 5 + C2 - Sin ( x ) Sin(x ] Sin(x ) = 5 ex sin ( x ) - 5 2 . ex cos ( x ) + CIcos(x ) + ( 2 sin ( x )a Solve the following differential equation (nd the general solution) using the method of \"reduction of order". Show all your work. You must. use reduction of order - if you use any other method. you will get no credit. (10 points) You cannot use any formula for the reduction of order. You must do the way we did in the class. Make sure to review and rewatch the recorded video on this topic before proceeding. If you use any formula, you will get 0