Need clear steps Thank you so much This is a typical pursuit

Need clear steps. Thank you so much.

This is a typical pursuit problem. A squirrel at (1,0) notices Rover at (0,0) and takes off at time t o seconds in the direction parallel to the xis at a speed b units per second. Rover at the same instant notices the squirrel and takes off in pursuit at speed a units per second, where a b. Rover chases the squirrel, i e. he a ways runs directly toward the current location of the squirrel. What is the curve that Rover follows and when and where does he catch the squirrel? dy bt a. At time t, the squirrel is located at (1, tb and Rover is located at (x(t), y(t)). Since Rover is chasing the squirrel, the differential equation must be satisfied because Rover\'s tangent line must intersect the squirrel. Yes, Rover ignores the speed of light.) Unfortunately the right hand side of the differential equation depends on both x and t. (If it depended only on x, we could solve for y and be done.) ds dx dt b. But, there is another differential equation. Rover moves at constant speed a and so the derivative of his arclength s 11 +(y is a 1 (y! Having a system of two and so dx dt dt dt differential equations, we solve it by eliminating variables to get a single equation. We know and so we differentiate the first equation so that it contains this quantity. Multiplying both sides of the first equation by (1 -x) and dt dt differentiating gives (1 -x)y -y Jba -y Substituting the value of one gets (1 -x)y c. This is a non-linear differential equation of order 2 But, notice that the equation contains only y and y but not y. So, if you think of y as the dependent variable then the equation is of order 1 and as such is separable and we have the initial conditions y (0) and x (0) d. The solution of this initial value problem is y an expression in x, a, and b). (To get this answer, you have to solve the implicit solution to get an explicit expression for y e. To finish off, you just need to integrate to get y using the initial condition to get rid of the constant of integration. The final result is y( an expression in x, a, and b) f Rover catches the squirrel at the point (1, d) where d an expression in a and b) and this happens at time t

Solution

y = x

and we

y can be solved as you see that y\'= u y\'\' = u

y can be solved as you see that y\'= u y\'\' = uy can be solved as you see that y\'= u y\'\' = uy can be solved as you see that y\'= u y\'\' = u