Evaluate `(2x+1)^2(d^2y)/(dx^2)-2(2x+1)(dy)/(dx)-12y=6x`
Solution
`(2x+1)^2(d^2y)/(dx^2)-2(2x+1)(dy)/(dx)-12y=6x`
Put `(2x+1)=e^z => x=(e^x-1)/2`
`(dz)/(dx)=2/(2x+1)` but`(dy)/(dx)=(dy)/(dx)(dz)/(dx)=2(dy)/(dx)=2/(2x+1)"Dy" "where" "D"=d/(dz)`
`therefore(2x+1)(dy)/(dx)=2"Dy"`
`therefore(2x+1)^2(d^2y)/(dx^2)=2^2"D(D-1)y"`
From (1),
`4D(D-1)y-4Dy-12y=6((e^x-1)/2)`
`(4D^2-8D-12)y=3(e^z-1)`
For complementary solution ,
`(4D^2-8D-12)=0`
∴D = -1,3
`thereforey_c=c_1e^(-z)+c_2 e^(3z)`
For particular integral ,
`y_p=1/(f(D))X`
`y_p=1/(4D^2-8D-12)(3(e^z-1))`
`therefore y_p=3/4 1/(D^2-2D-3)(e^z-1)` put D = a = 1 and D = a = 0
`thereforey_p=3/4(1/3-e^z/4)`
The general solution of given differential eqn is ,
`thereforey_g=y_c+y_p=c_1e^(-z)+c_2e^(3z)+3/4(1/3-e^z/4)`
Resubstituting 𝒛 ,
`therefore y_g=c_1(2x+1)^(-1)+c_2(2x+1)^3+3/4(1/3-(2x+1)/4)`
