MHB Solving Second Order ODE: True or False?

[Guest2]

I'm supposed determine whether following statements are true or false. However, I can't get past the notation.

Question: the second order differential equation $\ddot{x}+\dot{x}+x = 9t$ is:

(a) equivalent to $\begin{cases} \dot{x} = y, & \\ \dot{y}=-y-x+9t, &\end{cases}$ (b) solved by integrating factors, (c) linear.

I don't understand what the dots mean to properly do the question. So what do you they mean? I'd also appreciate any help on the question itself.

 

[Ackbach]

It's a convention, using Newton's notation, that primes denote derivatives w.r.t. $x$, or space, and dots represent derivatives w.r.t. $t$, or time. So your DE is equivalent to
$$\frac{d^2x(t)}{dt^2}+\d{x(t)}{t}+x(t)=9t,$$
where I have written $x=x(t)$ to emphasize that we are thinking of $x$ as a function of time. As for seeing if a is true, try differentiating the first equation, and substituting into the second, to see if you can recover the original DE. I'm not sure about b, but c is definitely true.

 

[MarkFL]

I agree it is linear, and in its original form, I would solve it using the method of undetermined coefficients, but you could also use the annihilator method, or variation of parameters. :)