Hello I have been having trouble with the differential equations solving applications. For instance on the D.E. 4y’‘-4y’ +y = e^(x/2)sqrt(1-x^2), The step by step solution seems to have it correct until it simplifies Yp(X). 4y''-4y'+y = e^(x/2)sqrt(1-x^2) - Wolfram|Alpha Here is the link to the equation. How does YpX simplify that dramatically? I even typed the simplified equation in question right into the wolfram alpha engine and it simplified it differently. (It simplified it exactly the way i did it by hand) Therefore is there a step missing in the step by step solution? Or is there something wrong with the algorithm? I will humbly accept any mathematical criticism for being wrong. I am not saying that Wolfram Alpha is wrong, (most likely myself) however It would be nice if someone would look at it.
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:
I guess you may have overlooked the simplified form of Sqrt[1-x^2]^(3/2).