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.

# Calculus Problem (Differential Equations): Solve the first-order differential eq

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