# Application of ODE

I got to do some assignment for my Student Research. 1. Cat and mouse are in room which is shape in sqare . Cat is in one topic square, mouse in another one. In topic square oposite than cat position is hole. Mouse and Cat are runing in same time to the hole. Cat is 2x faster than mouse. Will cat catch mouse before mouse escape ?

$$y = y(x)$$ - equation for curve (cat's runing ). That curve has feature that tangent in any points at the intersection with the right $$x = 4$$ (that right is mouse running) give coordinates of mouse. How cat is 2x faster, we got that Cat makes 2x longer way. In the moment $$x_0$$, cat made way $$L(x_0) = \int^{x_0}_0\mathrm ds\sqrt{1 + [y'(s)]^2}$$.

Find position of mouse in same moment $$x_0$$. How mouse position is on tangent of curve $$y=y(x)$$, in point $$(x_0,y(x_0))$$ we have that is equation of that tangent is $$p(x)-y(x_0) = y'(x_0)(x-x_0)$$.

So cordinates of mouse in that moment are: $$(4,y(x_0)+y'(x_0)(4-x_0))$$, from where we get an integral equation $$\int^{x}_0\mathrm ds\sqrt{1+[y'(s)]^2}= 2[y(x)+y'(x)(4-x)]$$.

From this we get D.E $$2\sqrt{\sqrt{1+[y'(s)]^2}} = 2(4-x)y''(x)$$ and started conditions $$y(0)=0$$ and $$y'(0)=0$$.

When we solve this and plot we get that Cat will cath mouse at $$8/3$$ metres.

My question is how I can solve this with as many as possible Mathematica and how I can on the end on the best way Plot this, and can I make some manipulate plot which will show positions and moving cat-mouse?

• Have a look at DSolve/DSolveValue and NDSolve/NDSolveValue. And ParametricPlot for plotting. – Henrik Schumacher Nov 18 '18 at 17:45
• Yes but thats not simple, you need more practice to make that – Милош Вучковић Nov 18 '18 at 17:46
• Yeah. Practice is something that nobody but yourself can give you. – Henrik Schumacher Nov 18 '18 at 17:47
• Have you made any attempt to solve this yourself? Where did you get stuck? – Chris K Nov 19 '18 at 13:14
• I did , DSolve that last second order D.E etc and i asked how i can make manipulate plot and etc, too see visual movements of cat and mouse – Милош Вучковић Nov 19 '18 at 13:42