Circular motion of a car

A model racing car of mass 0.2 kg is attached to a model string of length 10m. The car is moving anti clockwise ... See attached file for full problem description.

Solving a homogeneous differential equation

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Solving 2 ODE's with one unknown/algebra

Please see attachment, and use equation editor as I did so there is no confusion in your solution. Thank you! --- (See attached file for full problem description)

Differential equations, The Contraction Mapping Theorem

1). Define T : C[0,1] --> C[,1] by (Tx)(t) = 1 + integral from 0 to 1 x(s)ds. Is T a contraction? ( Please justify every step and claim, I want a proof not a yes or no only). P. S. I believe C[0,1] is the set of all the continuous functions on [0,1]. 2). Consider the operator in C[0,1], Ty(t) = integral from 0 to t (t-s)*y(s)ds. Show that T is a contraction. ( Also here I want a detailed proof).

Set of functions defined on [0,1] that have a continuous derivative there ( one-sided derivatives at the endpoints).

A). Let M be the set of functions defined on [0,1] that have a continuous derivative there ( one-sided derivatives at the endpoints). Let p(x,y) = max_[0,1]|x'(t) - y'(t)|. 1).Show that ( M,p) fails to be a metric space. 2). Let p(x,y) = |x(0) - y(0)| + max_[0,1]|x'(t) - y'(t)|. Is (M,p) now a metric space? Please justify all your answers..I want proofs here not a yes or no answers. ----------------------------------------------------------- B). Let M be the set of continuous functions on [0,1] and define p(x,y) = integral from 0 to 1 of |x(t) - y(t)|dt. Does this define a metric space? ( Also a proof here please for the yes or no answer).