### Separation of Variables : Separating PDEs into two or three ODEs (5 Problems)

Separation of Variables... Please see the attached file for the fully formatted problems.

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Separation of Variables... Please see the attached file for the fully formatted problems.

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(

Draw the phase portrait for the following system: x' = xy y' = (x^2)y

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.

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

Draw the phase portrait associated with x'' - 2x' + 2x = 0 and draw a rough sketch of the solution x(t) that satisfies x(0) = 1 and x'(0) = 0.

Convert {see attachment} into a system of differential equations and classify the resulting system

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.

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

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

Find a general solution to the system: x1' = -3x1 + x2 x2' = -4x1 + 2x2

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 ?

Determine whether the given initial value problem has a unique solution (using Theorem 1). *Please see attachment for problem (Direct "yes/no" answer is fine)

Find the general solution of the ODE below: y" + 2y' + 101y = 0

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

Solve the linear Differential Equation (see attachment) y'-y=exp(2x) y(0)=0 y"+6y'+10y=0 2yy'=1-y^2 y(0)=-2

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 {see attachment} for y(x) using the change of variables z = y + x.

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)

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

Find particular solution to differential equation 3y'' + 4y' + 1y = 1t^2 -2t + 2e^(-3t)

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

Find solution of: y'' - 2y' + y = 16exp((5)t) y(0)=1 y'(0)=6

Find y as a function of x if (x^2)(y'')-7xy'-9y = x^2 y(1)=6 y'(1)=8

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.

Please see the attached file for full problem description. --- Here is the problem: Find the center of mass of the region bounded by the parabola y = 8 -2x^2 and the x-axis a) if the density lambda is constant and b) if the density lambda = 3y

In order to solve differential equations, it is helpful to classify them as belonging to one or more categories. In this entry we will consider three common classes of first order ordinary differential equations (ODEs): separable, exact and linear. We will show how each class is defined.

This question is part of a study guide for my final test. Solve the following initial value problem. (see attached for problem) Thank you

(d^2 * y)/(d * t^2) + 6 * (dy/dt) + 9y = 0 y(0) = 10, y'(0) = 0

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