The image shows a math exam with various problems. Here's a breakdown of each question for better understanding and solving.

Problem 1

Find the critical points and classify them (local maximum, local minimum, or saddle point) for the function:

$f(x,y) = x^2 + y^2 + 2xy$

Problem 2

Find the solution to the following differential equation:

$y' = \frac{1}{x} \left( y + \frac{1}{y} \right)$

Problem 3

Evaluate the double integral:

$\iint_{D} (x+y) \, dx\,dy, \quad D = \{(x,y) \, | \, 0 \leq x \leq 1, \, 0 \leq y \leq 1 - x\}$

Problem 4

Solve the differential equation:

$y'' + \left( x^2 - 4 \right)y = 0$

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