2021-043

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

$$y^{\prime }-2xy=x$$

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.

$$x{f}^{\prime }-2f=x^3\sin x$$

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

Original idea by: Thales Almeida