Problem $1$:

Verify that $y=e^{3x}cos(2x)$

is a solution of the given differential equation :

$y^{\text{'}\text{'}}-6y^{\text{'}}+13y=0$

Problem $2$:

Verify that $y=\frac{1}{(1+c_1{e}^{-x})}$

is a solution of the given differential equation:

$y^{\text{'}}=y-y^2$