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Q)

Find the differential equation whose general solution is $y= Ae^x+Be^2x+Ce^3x+De^4x+E$ where A,B,C,D,E are constant

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A)
Solution :
$y= Ae^{x}+Be^{2x}+ce^{3x}+De^{4x}+E$ -------------(1)
$y^1= Ae^{x}+2Be^{2x}+3ce^{3x}+4De^{4x}$ -------------(2)
$y^1-y = Be^{2x} +2Ce^{3x} +3De^{4x}$ --------------(3)
Differentiating equation (3) w.r.t.x
$y^{11}-y^1=2Be^{2x} +6ce^{3x} -12 De^{4x}$ ------------------(4)
Multiply equation (3) by 2 and subtract from (4)
$y^{11} - 3y^{1} +2y=2ce^{3x} +6 De^{4x}$ -----------------(5)
Difference equation (5) w.r.t.x
$y^{111} - 3y^{11} +2y^{1}=6ce^{3x} +24De^{4x}$ -----------------(6)
Multiply equation(5) w.r.t.x by 3 and subtract from (6)
$y^{111} - 6y^{11} +11y^{1}-6y=6De^{4x}$ -----------------(7)
Differentiate equation (7) w.r.t.x
$y^{111} - 6y^{111} +11y^{11}-6y^{1}=24 De^{4x}$ -----------------(8)
Multiply equation (7) by 4 and subtract from (8)
$y^{111} +10y^{11} +35y^{11}-10y^{1}-24y=0$
It is the required differential equation .