Investigate the asymptotic behaviour of the solutions $y$ of the equation $x^5 + x^2y^2 = y^6$ that tend to zero as $x\to0$.
To investigate the asymptotic behavior of the solutions \( y \) of the equation \( x^5 + x^2 y^2 = y^6 \) that tend to zero as \( x \to 0 \), we will assume that \( y \) behaves like \( x^a \) for some exponent \( a \). Specifically, we assume \( y = x^a (c + o(1)) \) as \( x \to 0 \), where \( c \) is a constant and \( o(1) \) represents terms that go to zero faster than \( x^a \).

1. **Substitute \( y = x^a (c + o(1)) \) into the equation:**
   \[
   x^5 + x^2 (x^a (c + o(1)))^2 = (x^a (c + o(1)))^6
   \]
   Simplifying each term, we get:
   \[
   x^5 + x^2 x^{2a} (c^2 + o(1)) = x^{6a} (c^6 + o(1))
   \]
   \[
   x^5 + x^{2 + 2a} (c^2 + o(1)) = x^{6a} (c^6 + o(1))
   \]

2. **Analyze the terms to find the dominant balance:**
   For the asymptotics to hold, at least two of the terms must cancel each other out up to the leading order, and the lower-order terms in the canceling pair must dominate the remaining term as \( x \to 0 \).

   - **Case 1: \( x^5 \) cancels with \( x^{2 + 2a} \):**
     \[
     x^5 = x^{2 + 2a} \implies 5 = 2 + 2a \implies a = \frac{3}{2}
     \]
     Substituting \( a = \frac{3}{2} \) into the equation:
     \[
     x^5 + x^{2 + 2 \cdot \frac{3}{2}} (c^2 + o(1)) = x^{6 \cdot \frac{3}{2}} (c^6 + o(1))
     \]
     \[
     x^5 + x^5 (c^2 + o(1)) = x^9 (c^6 + o(1))
     \]
     For the terms to cancel, we need:
     \[
     1 + c^2 = 0 \implies c^2 = -1
     \]
     Since \( c \) must be real, this case has no solution.

   - **Case 2: \( x^5 \) cancels with \( x^{6a} \):**
     \[
     x^5 = x^{6a} \implies 5 = 6a \implies a = \frac{5}{6}
     \]
     Substituting \( a = \frac{5}{6} \) into the equation:
     \[
     x^5 + x^{2 + 2 \cdot \frac{5}{6}} (c^2 + o(1)) = x^{6 \cdot \frac{5}{6}} (c^6 + o(1))
     \]
     \[
     x^5 + x^{\frac{22}{6}} (c^2 + o(1)) = x^5 (c^6 + o(1))
     \]
     The term \( x^{\frac{22}{6}} = x^{\frac{11}{3}} \) dominates the other terms as \( x \to 0 \), so this is a spurious solution.

   - **Case 3: \( x^{6a} \) cancels with \( x^{2 + 2a} \):**
     \[
     x^{6a} = x^{2 + 2a} \implies 6a = 2 + 2a \implies 4a = 2 \implies a = \frac{1}{2}
     \]
     Substituting \( a = \frac{1}{2} \) into the equation:
     \[
     x^5 + x^{2 + 2 \cdot \frac{1}{2}} (c^2 + o(1)) = x^{6 \cdot \frac{1}{2}} (c^6 + o(1))
     \]
     \[
     x^5 + x^3 (c^2 + o(1)) = x^3 (c^6 + o(1))
     \]
     For the terms to cancel, we need:
     \[
     c^2 = c^6 \implies c^2 (1 - c^4) = 0
     \]
     This gives \( c = \pm 1 \). The \( x^5 \) term is dominated by the \( x^3 \) terms as needed.

Therefore, the asymptotic behavior of the solutions is:
\[
y = \pm x^{1/2} + o(x^{1/2})
\]

The final answer is \( \boxed{ y = \pm x^{1/2} + o(x^{1/2}) } \)