There is no predicate in
"Prove that the integration
especially where y= x if R+ SX= 0"
Prove the the integeration is what? And it's not clear what "R+ SX= 0" means. Is "X" the same as "x"? If not, what is it? If so, R+ Sx= 0 is true for just one value of x, specifically x= -R/S.
The second equation can be written as by multiplying both sides by . And then the change of variable x= ln(t) gives an equation "with constant coefficients" like the first equation.
You are now saying, since "X= Y", that you must prove that R+ YS= 0, which is the same as Y= -R/S, satisfies the equation .
That makes sense now, but is wrong! If you replace Y by the constant function -R/S in that equation, and DY are both 0 since Y is a constant and the equation reduces to SY= -R= 0 which is not generally true.
Perhaps you meant "Show that Y= R+ XS, where X is the independent variable, satisfies ."
Now, Dy= D(R+ XS)= S and so the equation becomes 0+ R(S)+ (R+XS)(S) which is still not 0.
I still have no idea what you are really asking.