Explicit Euler Method

Was Euler the ancient fortune-teller? He almost was. One of his principles, the explicit method says that your future days could be predicted from your present day knowing your past provided your time frame is not too large. Have a look at the solution to a system of non-linear differential equations system using explicit or ...continues

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

Show that the solution of: dM/dt = Poert – pM M(0)=0 is M(t) = [Po/(r + p)](ert – e-pt) Please see the attached file for the fully formatted problems.

Solving 2 ODE's with One Unknown

Given that: dMN/dt = P¬oert – pMN + m(MS – MN) dMS/dt = m(MN - MS) – pMS¬ MN¬ (0) = MS (0) = 0 And using: M¬N + MS = [Po/(r + p)](ert – e-pt) Show that MS(t) = And show that R = MN(t) / MS(t) = Please see the attached file for the fully formatted problems.

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)* ...continues

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 ...continues

Finding a particular solution of a differential equation

I already solved the homogeneous portion, and I need help solving the particular solution and of course combining the two to get the entire solution to the differential equation. Not too difficult - see attachment. Please use equation editor if possible. Thank you. --- Given that: dMS/dt = m(MN - MS) - pMS¬ ...continues

Solving a Particular solution to an Ordinary Differential equation

See attached problem. PLEASE NOTE!!! I have noted in the problem statement that I have solved the homogeneous portion of the differential equation, and I need assistance in solving for the particular solution and finally the whole solution. I have gotten 2 responses from other OTA's that are as follows: "The point is that y ...continues

Local or Uniform Lipschitz Constants

Determine if the following functions satisfy local or uniform Lipschitz condition. 1). te^y My work: I found d/dy (te^y) = te^y, and this is not bounded above for any value of y, so this made me conclude that it has locally Lipschitz condition since the Lipschitz constant here changes as the reagion changes? Am I right? ...continues

Initial-Value Problem for System of Differential Equations : Fundamental Theorem of Calculus

Let Q(t) =< (less than or equal) C + integral from t_0 to t ( K(s) Q(s) ) ds, Where Q(t) is a nonegative function , C > 0 and K(s) >= 0. a).Show that: Q(t) =< Ce^( integral from t_0 to t ( K(s)ds) ), t >= t_0 b). What conclusion can be made if C = 0? ( Note that proof in a may fail is C = 0 ). I want a det ...continues