How to find the power series solution of this ordinary differential equation using MMA

Question

I already know that the solution of this differential equation $y''(x) - x*y(x) = 0$ can be expressed by the following power series:

$$y(x)=c0(1+\frac{x^{3}}{2\times3}+\frac{x^{6}}{2\times3\times5\times6}+\cdots\dfrac{x^{3n}}{2\times3\times5\times6\times\cdots\times(3n-1)3n}+\cdots)+c1(x+\frac{x^{4}}{3\times4}+\frac{x^{7}}{3\times4\times6\times7}+\cdots\dfrac{x^{3n+1}}{3\times4\times6\times7\times\cdots\times(3n)(3n+1)}+\cdots)=c0(1+\sum _{i=1}^\infty \frac{x^{3 i}}{\prod _{j=1}^i (3 j-1) (3 j)})+c1(x+\sum _{i=1}^\infty \frac{x^{3 i+1}}{\prod _{j=1}^i (3 j) (3 j+1)})$$

In the above formula, c0 and c1 are arbitrary constants.

I want to know how to use MMA to find the power series solution of the above ordinary differential equation and the power series solution of the following differential equation: $$y'(x)^2=\frac{9}{y(x)}-1$$

Other examples for testing:

DSolve[{y[x]*y''[x] == 1 + y'[x]^2, y[0] == 1, y'[0] == 0}, y[x], x]

I also want to get the infinite power series solution of this above differential equation, but I haven't solved this problem yet.

Answer 1

For the first one, expansion around zero

ode = y''[x] - x y[x] == 0
AsymptoticDSolveValue[ode, y[x], {x, 0, 10}]

For second one

ode = (y'[x])^2 == 9/y[x] - 1
AsymptoticDSolveValue[ode, y[x], {x, 0, 10}]

with warning:

AsymptoticDSolveValue::asdb There are multiple solution 
  branches for the equations, but AsymptoticDSolveValue will return only

one

See help on symptoticDSolveValue for more info. This is new function added in 11.3