Verify that \(y=\cos x\) solves \(y''+y=0\).

Solve \(y''=6x\).

Solve \(y''-5y'+6y=0\).

Solve \(y''+4y=0\).

Solve \(y''-4y'+4y=0\).

Solve the initial value problem \(y''+y=0\), \(y(0)=2\), \(y'(0)=3\).

Find the characteristic equation for \(2y''+3y'-5y=0\).

Solve \(y''+2y'+5y=0\).

Find a particular solution to \(y''-y=e^x\) using the trial form \(y_p=Ax e^x\).

For \(y''+9y=0\), state the general solution and identify the oscillation frequency in the argument of sine and cosine.