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SolnSolution to Linear second Order Homogeneous DES Using the Method of Reduction of order rarr{:y^('')+p(x)y^(')+theta(x)y=0quad" ( ")Given y_(1)(x) is a solution, then the second solution y_(2)(x) isy_(2)(x)=y_(1)(x)int(e^(-int p(x)dx))/(y_(1)^(2)(x))dx" (2) "Here given DE is {:[(2x+1)y^('')+2(4x+1)y^(')+8xy=0],[y^('')+(2(4x+1))/((2x+1))y^(')+(8x)/(((2x+1))/(2x+1)y=0)],[" [divided by "(2x+1)]:}comparing (1) & (3)P(x)=(2(4x+1))/((2x+1)),quad theta(x)=(8x)/((2x+1))and also given y_(1)=e^(-2x) (5)Now put value from (4) < (5) in (2){:[y_(2)(x)=e^(-2x)int(e^(-int(2(4x+1))/((2x+1))dx))/((e^(-2x))^(2))dx],[=>y_(2)(x)=e^(-2x)int(e^(-int(8x+2)/(2x+1)dx))/(e^(-4x))dx],[[:'(c^(a))^(b)=e^(ab):}]:}{:[=e^(-2x)int(e^(-4x+ln(2x+1)))/(e^(-4x))dx],[=e^(-2x)int(e^(-4x)e^(ln(2x+1)))/(e^(-4x))dx],[=e^(-2 ... See the full answer