2021-043

Choose the alternative that provides a solution for the following differential equation (assume that C is a constant):

$$ y' - 2 x y = x $$

1. \( 2 + 4x^2 \)
2. \( 4x^2 - x^3 \)
3. \( C e^{x^2} - 1/2 \)
4. \( C e^{3x} + 2 \)
5. None of the above

Original idea by: Thales Almeida

2021-042

Consider a function \( f(x) \) given by:

$$ f(x) = - x^\alpha \cos x + C x^\alpha . $$

Choose the value of \( \alpha \) that makes the following formula true.

$$ x f' - 2 f = x^3 \sin x $$

1. \( \alpha = 0 \)
2. \( \alpha = 1 \)
3. \( \alpha = 2 \)
4. \( \alpha = 3 \)
5. None of the above.

Original idea by: Thales Almeida