If mouse moves along a circle and the cat is not on the circle, the path of the cat would be a spiral approaching a circle. If the lag time between successive movements goes to zero the nearer to a circle the cat would move. Where is your lag time represented or suddenly it is instantaneous?
Instantaneous
The equation you came up with is a PDE, not an ODE. To reduce a variable you could try looking into Laplace Transforms of PDE. Also, you used both equations to arrive at your simplification, but you still lost information by only displaying one equation. You should keep it as a system, probably with (x')^2 + (y')^2 = 1.
I may be wrong, but it sounds like you're a graduate student or an advanced undergraduate student. Why don't you talk about this with your professor?
Oh, and use \begin{equation} \end{equation} rather than $$ $$ in your work. The double dollar signs are outdated and produce spacing problems.
The equation you came up with is a PDE, not an ODE. To reduce a variable you could try looking into Laplace Transforms of PDE. Also, you used both equations to arrive at your simplification, but you still lost information by only displaying one equation. You should keep it as a system, probably with (x')^2 + (y')^2 = 1.
I may be wrong, but it sounds like you're a graduate student or an advanced undergraduate student. Why don't you talk about this with your professor?
I am 16 years old and for the past year and a half I have been teaching myself this content from textbooks I find at my library (maybe second or third year IDK). I used to be able to talk to my high school teachers about the problems however I have surpassed them. I usually send these unsolved problems to a mentor of mine, Ludvik Bass an emeritus professor of the university of Queensland but due to Covid he is self isolating.
Also thank you for the suggestions.
Well, most 16 year-olds don't use LaTeX. I'm impressed!
But since you haven't had professors teach you PDE, then I will tell you a general concept: PDEs are hard. For the most part you won't be able to come up with an explicit solution, even in terms of an integral. For applications people use numerical methods to approximate what is going to happen in the short term and analytical methods to predict asymptotic behavior, i.e. what is going to happen in the long run.
Well, most 16 year-olds don't use LaTeX. I'm impressed!
But since you haven't had professors teach you PDE, then I will tell you a general concept: PDEs are hard. For the most part you won't be able to come up with an explicit solution, even in terms of an integral. For applications people use numerical methods to approximate what is going to happen in the short term and analytical methods to predict asymptotic behavior, i.e. what is going to happen in the long run.
@SVUDrBell Would you recommend studying non-linear dynamics and chaos theory prior to PDEs to get a better idea of differential equations? And isn't the DE ordinary since it only has time as a variable?
I am currently studying complex ODEs and became interested in pursuit curves. However I am having trouble simplifying the equations into solvable DEs. Here is my work:
If mouse (m) moves along a curve then cat (c) describes the pursuit curve which minimises the path following the movement of the mouse. In the following problem for simplicity the cat’s velocity (c’) will equal 1 and be parallel to m-c. Thus, these equation are formed.
$$m=\langle u(t),v(t)\rangle ,c=\langle x(t),y(t)\rangle$$
$$\frac{m-c}{ \| m-c\| } . \frac{c}{ \| c\| } = \frac{(u-x) x' +(v-y) y' }{ \sqrt{(u-x)^{2}+(v-y)^{2}} } =1$$
$$ (x')^{2}+(y')^2 =1$$
Manipulating the first equation yields the following
$$ (u-x)x'+(v-y)y'=\sqrt{(u-x)^{2}+(v-y)^{2}}$$
$$\ \big((u-x)x'+(v-y)y'\big)^2-(u-x)^2-(v-y)^2=0 $$
$$(u-x)^2 (x')^2+2(u-x)x'(v-y)y'+(v-y)^2(y')^2-(u-x)^2-(v-y)^2=0$$
$$(u-x)^2\big((x')^2-1\big)+(v-y)^2\big((y')^2-1\big)+2(u-x)x'(v-y)y'=0$$
$$-(u-x)^2(y')^2-(v-y)(x')^2+2(u-x)x'(v-y)y'=0$$
$$[(u-x)y'-(v-y)x']^2=0 \Rightarrow (u-x)y'-(v-y)x'=0$$