Find the values of A for which the given function solves the given ODE:

$($a$)5y^{\text{'}}=2y$ and $y=e^{Ax},$

$($b$)2y^{\text{'}\text{'}}+7y^{\text{'}}-4y=0$ and $y=e^{Ax},$

$($c$)x^2{y}^{\text{'}\text{'}}-7x{y}^{\text{'}}+15y=0$ and $y=x^A,$

$($d$)\frac{dy}{dt}+20y=24$ and $y=A-\frac{6}{5}{e}^{-20t},$

$($e$)y^{\text{'}\text{'}}+y=tan(x)$ and $y=$Acos$(x)ln(sec(x)+tan(x)).$

$($f$)2x{y}^{\text{'}}-y=$Axcos$(x)$ and $y=\sqrt[\phantom{2}]{}n{t}_1^x\frac{cos(t)}{\sqrt[\phantom{2}]{t}}dt,$

$($g$)y^{\text{'}}+2xy=1$ and $y=A(e^{-x^2}+e^{-x^2}{\displaystyle \underset{0}{\overset{x}{}}}{e}^{t^2}dt),$

$1$