# How to solve the differential equation

I need to solve the differential equation

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

How to solve it correctly?

• 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.
– Bill
Jul 17 '20 at 4:26

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]];


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


Simplify[sol - proposedSolution]


• 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? Jul 17 '20 at 7:56
• @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. Jul 17 '20 at 11:01