I have a problem with solving the following differential equation: $(x^a u'(x))'=0$, where x and a are in $(0,1)$ and $[0,1)$, respectively.The derivative given in equation is with respect to $x$. I used Mathematica and got the following solution: $C_{2}+ (C_{1}/1-a)*x^(1-a)$. But I didn't get it's detail. Could you please share detail for this solution or help me in getting step by step solution from Mathematica?
All the best,
Clear["Global`*"]
eqn = D[x^a u'[x], x] == 0
(* a x^(-1 + a) u'[x] + x^a u''[x] == 0 *)
sol = DSolve[eqn, u, x][[1]]
(* {u -> Function[{x}, (x^(1 - a) C[1])/(1 - a) + C[2]]} *)
Verifying the solution
eqn /. sol
(* True *)
u[x] /. sol
(* (x^(1 - a) C[1])/(1 - a) + C[2] *)
Mathematica doesn't provide step-by-step solutions. However, it can be used to call WolframAlpha which does in some cases -- such as this
WolframAlpha["DSolve[a x^(a-1) u'[x] + x^a u''[x] == 0, u[x], x]"]
Editwindow, There is a{ ... }icon to format it as code. $\endgroup$