Second-order linear ODEs

Are $\sin(x)$ and $e^x$ linearly independent? Justify.

Are $e^x$ and $e^{x+2}$ linearly independent? Justify.

Guess a solution to $y'' + y' + y= 5$.

Find the general solution to $x y'' + y' = 0$. Hint: It is a first-order ODE in $y'$.

Write down an equation (guess) for which we have the solutions $e^x$ and $e^{2x}$. Hint: Try an equation of the form $y''+Ay'+By = 0$ for constants $A$ and $B$, plug in both $e^x$ and $e^{2x}$ and solve for $A$ and $B$.