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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
    Dec 18, 2022 at 5:25

1 Answer 1

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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]"]

enter image description here

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