# Solving differential equation

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,

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– Syed
Commented Dec 18, 2022 at 5:25

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]"]
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