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Starting with the equation

$$y^{\prime\prime}x^2={y^\prime}^2$$

I could only get the general solution using DSolve:

DSolve[{y''[x] x^2 == y'[x]^2}, y[x], x]
(* {{y[x] -> -(x/C[1]) + C[2] - Log[1 - x C[1]]/C[1]^2}} *)

However, I failed to get the singular solutions y[x] -> x^2/2 + C[1] and y[x] -> C[1]. So, how can I get them?

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2 Answers 2

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With :

sol = DSolve[{x^2 y''[x] == y'[x]^2}, y[x], x][[1, 1, 2]];

Limit[sol, C[1] -> 0]
(* x^2/2 + C[2] *)

Limit[sol, C[1] -> Infinity]
(* C[2] *)
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I failed to get the singular solutions

As of V 13.1, there is now an option to DSolve to find singular solutions.

ode = y''[x]  x^2 == y'[x]^2
DSolve[ode, y[x], x, IncludeSingularSolutions -> True]

Gives

enter image description here

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