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?

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]