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I need to solve the differential equation

enter image description here

DSolve[{(1-x)y''[x]==1/5 \[Sqrt](1+y'[x]^2),y[0]==0,y'[0]==0},y[x],x]

but the result given seems to be incorrect and returns an error "Inverse functions are being used by Solve, so some solutions may not be found; use Reduce for complete solution information". How to solve? The correct solution should be

enter image description here

How to solve it correctly?

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  • $\begingroup$ Look carefully at Rationalize[N[y[x]/.DSolve[...]]] and decide whether "the result given seems to be incorrect." As always with Mathematica, there are probably half a dozen other ways of doing anything and there may be a better way of doing that. Also, errors are very different from warnings, but a new user may not be able to tell the difference. $\endgroup$
    – Bill
    Jul 17, 2020 at 4:26

1 Answer 1

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ClearAll[x,y];
ode = (1 - x) y''[x] == 1/5 Sqrt[1 + y'[x]^2];
ic = {y[0] == 0, y'[0] == 0};
sol = y[x] /. First@DSolve[{ode, ic}, y[x], x];
sol = Assuming[x < 1, FullSimplify[sol]];

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proposedSolution = -5/8*(1 - x)^(4/5) + 5/12 (1 - x)^(6/5) + 5/24;

Mathematica graphics

Simplify[sol - proposedSolution]

Mathematica graphics

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  • $\begingroup$ There is a warning "Solve::ifun: Inverse functions are being used by Solve, so some solutions may not be found; use Reduce for complete solution information." Why? $\endgroup$
    – Mertin
    Jul 17, 2020 at 7:56
  • $\begingroup$ @Mertin because it could not solve it, it needs assumptions which are missing. DSolve does not take assumptions, so you have to do that afterwords. Solution you posted is valid for $x<1$ only. $\endgroup$
    – Nasser
    Jul 17, 2020 at 11:01

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