I would like to ask about the uniqueness of solutions to the ODE:
$$ay(x)y'(x)+by(x)+c=0$$
Given a condition $y(0)=k$.
Contextual motivation:
A vehicle of mass $M$ kilograms is travelling in a straight line on a horizontal road. Its engine develops a constant $P$ watts of power and the vehicle experiences a constant $R$ Newtons of resistance to its motion.
This sort of question appears frequently in the UK Further Maths A-Level mechanics. I wondered the other day if it were possible to solve for the motion of the vehicle exactly. That is, we are often asked questions of the form "If it takes the vehicle $30$ seconds to pass from point $A$ to point $B$ and if it [so fast] at $A$ and [so fast] at $B$ ..." but these question always require the exam board to give us a lot of information - since the acceleration is non-constant.
However, inspired by Wolfram's solution to the primitive equation $yy'+y+1=0$ (weirdly, Wolfram couldn't solve it in general) I figured out a way to solve for the vehicle's motion:
By Newton's second law and by $P=Fv$, the vehicle's motion is described by $Ma(t)=\frac{P}{v(t)}-R$ and some initial condition $v(0)$. We may re-write: $$\tag{$\ast$}Mv(t)v'(t)+Rv(t)-P=0$$After some experimentation with the ansatz solution $g(x)=W(e^x)+1$, where $W$ is the Lambert W/product log function, gives: $$\tag{1}v(t)=\frac{P}{R}\left(1+W\left(\lambda\exp\left(-\frac{R^2}{PM}t\right)\right)\right)$$Where $\lambda:=\left(\frac{R}{P}v(0)-1\right)\exp\left(\frac{R}{P}v(0)-1\right)$ satisfies the initial value condition.
Nicely enough for computer modelling, in any physical system we must have $P\gt Rv(0)$ for there to be positive acceleration in the first place, so that $|\lambda|\lt\frac{1}{e}$ and the term inside the $W$ function is less than $1/e$ for all positive $t$, so the power series for $W$ can be used. I have this setup in Desmos here. Finding explicit representations for $a(t)$ and $s(t)$ (displacement) as I have done in the graph is not so hard after some manipulation of the ODE $(\ast)$. It is also possible to invert $(1)$ for $T$, which I have done in the graph.
I haven't ever conducted a formal study of differential equations, so I really don't know any uniqueness theorems beyond the bog standard Peano and Picard theorems, which I don't think can be applied here. Is there another solution to $(\ast)$ other than $(1)$?
The reason I suspect there may be is due to this: I tested my model on two different mechanics questions from this exam board. In one of them, my model predicted their values precisely but in another of them my model gave very different values. That my model agreed with their question parameters for one question suggests the exam board puts time into ensuring physically accurate scenarios (rather than just throwing numbers at us, which would be fair enough considering we aren't expected to use an ODE solution), so the fact that my model differed on another question suggests that there could be other solutions.
N.B.:
The precise situation where my model didn't line up with the question was this:
$M:=1000,P:=15,000,R:=400$. The car arrives at point $A$ with velocity $10$ and arrives at point $B$ with velocity $20$. The journey takes $30$ seconds.
In my model, $T(20)=17.374$, not $30$. They could have just made $30$ up, but I am asking just in case they didn't.
Final note: my model is a valid solution - I have checked that all the functions in the Desmos graph satisfy $(\ast)$ and satisfy the basic relations, e.g. $T(V(x))=V(T(x))=x$ to high numerical accuracy (where $T$ is the function I am using to find the time at which the vehicle attains a certain velocity).