y\'\'\' + 3y\' -4y =e^(-2x) +5e^x Solution 1....1-x ?.....? [ (2x+3y)e^(x+y) ] d
ID: 1892229 • Letter: Y
Question
y''' + 3y' -4y =e^(-2x) +5e^xExplanation / Answer
1....1-x ?.....? [ (2x+3y)e^(x+y) ] dy dx -1...0 1-x ? [ (2x+3y)e^(x+y) ] dy ====> by part 0 u = (2x+3y) .........dv = e^(x+y) du = 3 dy..............v = e^(x+y) u * v - ? v * du (2x+3y) * e^(x+y) - ? e^(x+y) * 3 dy ......................................… (2x+3y) * e^(x+y) - 3e^(x+y) ] ......................................… [ ( 2x+3*(1-x) ) * e^( x+(1-x) ) - (2x+3*0) * e^(x+0) ] - [ 3e^(x+1-x) - 3e^(x+0) ] [ ( 2x+ 3 - 3x) ) * e^( 1) - (2x) * e^(x) ] - [ 3e^(1) - 3e^(x) ] ( -x + 3) ) * e^( 1) - (2x)e^(x) - 3e^(1) + 3e^(x) 1 ? [ ( -x + 3) ) * e^( 1) - (2x)e^(x) - 3e^(1) + 3e^(x) ] dx -1 1 ? [ -xe^( 1) + 3e^( 1) - (2x)e^(x) - 3e^(1) + 3e^(x) ] dx -1 1 ? [ -xe^( 1) - (2x)e^(x) + 3e^(x) ] dx -1 (2x)e^(x) =====> bu parts u = 2x .............dv = e^(x) du = 2 dx...........v = e^(x) u * v - ? v * du 2x * e^(x) - ? e^(x) * 2 dx 2x * e^(x) - 2e^(x) -(x^2e^( 1))/2 - [ 2x * e^(x) - 2e^(x) ] + 3e^(x) -[ x^2e^( 1) ] / 2 - 2x * e^(x) + 2e^(x) + 3e^(x) ......................................… -[ x^2e^( 1) ] / 2 - 2x * e^(x) + 5e^(x) ] ......................................… -[ (1)^2e^( 1) - (-1)^2e^( 1) ] / 2 - [ 2*1 * e^(1) - 2*-1 * e^(-1) ] + [ 5e^(1) - 5e^(-1) ] -[ 0 ] / 2 - [ 2e^(1) + 2/e^(1) ] + [ 5e^(1) - 5/e^(1) ] - [ 2e^(1) + 2/e^(1) ] + [ 5e^(1) - 5/e^(1) ] = 5.58
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