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Ordinary Differential Equations

Ordinary Differential Equations: Boundary Value

Solve each of the following ODE's for y(x): (a) y" + 16y = 0 with y(0) = 1 and y'(0) = 0. (b) y" + 6y' + 9y = 0 with y(0) = 1 and y'(0) = 0. (c) y"' - y" + y' - y = 0 with y(0) = 1 and y'(0) = y" (0) = 0.

Ordinary Differential Equation

Ordinary Differential Equation Determine if the following system has nay non-constant solutions that are bounded, i.e. do not run off to infinity in magnitude x' = x(y - 1) y' = y(

Show that this system is a Hamiltonian system.

Please give me a step-by-step solution to this attached ODE. Consider the following system: x' = -2y y' = x/2 a. Show that this system is a Hamiltonian system. b. Us a Hamiltonian to sketch the phase portrait for this system.

Space Factor, Time Factor, Eigenvalues and Sturm-Liouville

7. Consider the differential equation ut =1/2 uxx + ux for 0 <x < pi, t > 0 with boundary conditions u(0,t) = u(pi,t) = 0. (a) Separate variables and write the ordinary differential equations that the space factor X(x) and the time factor T(t) must satisfy. (b) Show that 0 is not an eigenvalue of the Sturm-Liouville proble

ODE: Variation of Parameters

Please see the attached file for the fully formatted problems. One solution of the equation attached is y(t) = t. Find the general solution. Use variation of parameters to find a particular solution of the equation attached.

Phase Plane

Solve y" + w^2y = 0 w= the symbol omega subscript zero (0).

Unique Equilibrium Levels : Lead Levels in Blood

For any positive values of the input I and the rate constants k, show that system (3) has a unique equilibtium solution x1=a, x2=b, x3=c where a,b and c are all positive. Building the Model ODEs Apply the Balance Law to the lead flow through the blood, tissue, and bone compartments diagrammed in Figure 61 .2 to obtain a syst

Determine Inertia, Damping, and Stiffness

See attached file. P228#2 Using the paradigm, What are the inertia, damping, and stiffness for the equation ? If y>0, what is the sign of the 'stiffness constant'? Does your answer help explain the runaway behavior of the solutions ?

First Order Ordinary Differential Equations

Problem A: Suppose that a giant HD-ready television of mass m falls from rest towards earth and its parachute opens at time t=0. when its speed is v(0)=v0 Since the TV is massive assume the drag force is proportional to the square of the velocity. Write a complete model for the velocity v(t) What is the asymptotic behavi

Mechanical displacement - Steel ball problem

A steel ball weighing 128 pounds (mass= 4 slugs) is suspended from a spring. This stretches the spring 128/485 feet. The ball is started in motion from the equilibrium position with a downward velocity of 9 feet per second. The air resistance (in pounds) of the moving ball numerically equals 4 times its velocity (in feet p

Solve a 2nd order ODE.

Use methods of undetermined coefficients to find one solution of: y'' + 2y' +2y = (10t+7)e^(-t)cos(t)+(11t+25)e^(-t)sin(t)

Solve a homogenous 2nd order ODE : Cauchy-Euler Equation

Find y as a function of x if: (x^2)(y'') + 19xy' +81y = x^2 y(1) = 9 y'(1) = -3 Hint: First assume that at least one solution to the corresponding homogeneous equation is of the form . You may have to use some other method to find the second solution to make a fundamental set of solutions. Then use one of the two metho

Evaluate: A Second Order ODE

Use the method of undetermined coefficients to find one solution of : y'' - 16y' +101y = 16exp(8t)cos(6t)+16exp(8t)sin(6t)+1*1

Ordinary first order differential equation

Find all solutions to the ODE yy'= (1-y^2) sin x. (When dividing by 1-y^2, be careful that you don't lose any solutions). NOTE: y2 = y squared Please see attached file for full problem description.

Second order ODE

Find y as a function of t if 64y'' + 32y' +4y = 0 y(5)=6 y'(5) = 5

Second order ODE

Find y as a function of t if 16y'' - 88y' +121y = 0 y(0) = 4 y'(0) = 9