Answer parts $($a$)($b$)($c$)$:

$($a$)$: Find all constant $($equilibrium$)$ solutions to the following ODE $($if there are none, write "none"$)$: $y^{\text{'}\text{'}}+e^{-t}{y}^{\text{'}}+25y=12sin(5t)$

$($b$)$ The brakes of a car are applied when it is moving at $12$ meters per second and provide a constant deceleration of $5$ meters per secondsecond ${?}^2.$Write an IVP $($ODE and initial condition$)$ for the velocity $v(t).$ That is$,$ it will be in the form:

$($c$)$: For each of the $5$ ODE's below, determine if it is linear or non$-$linear. List all of the linear ODE's in the answer box marked Linear ODE's", and list the nonlinear ODE's in the box marked Non$-$linear ODE's".

I$)y^{\text{'}}=2y$

II$)P^{\text{'}}=3P(1-P)$

III$)\frac{d^{2}f}{d{x}^2}+2\frac{df}{dx}-3f=0$

IV$)y^{\text{'}}+\frac{y}{2\sqrt[\phantom{2}]{t}}=\frac{1}{2\sqrt[\phantom{2}]{t}}$

V$)v^{\text{'}}=6\sqrt[\phantom{2}]{v}$