Are you referring to "The particular solution is thus: " to "Simplify"? The solution is correct by factoring out
$1/24 \times e^{x/2}$ then you just combine the terms with the same exponential or radical factors:
![radical](/c/portal/getImageAttachment?filename=ScreenShot2015-02-26at11.41.21PM.png&userId=23928)
I guess you may have overlooked the simplified form of Sqrt[1-x^2]^(3/2)
.