Calculus Problem (Differential Equations):
Solve the first-order differential equation
�
′
=
2
�
�
y
′
=2xy with the initial condition
�
(
0
)
=
1
y(0)=1. Find the solution
�
(
�
)
y(x) and describe its behavior as
�
x varies.

Algebra Problem (Linear Algebra):
Given a 3×3 matrix
�
A with eigenvalues
�
1
=
4
λ
1
​
=4,
�
2
=
−
1
λ
2
​
=−1, and
�
3
=
2
λ
3
​
=2, find the corresponding eigenvectors and determine whether the matrix is diagonalizable.