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Ordinary Differential Equations

An ordinary differential equation is an equation containing a function of one independent variable and its derivatives. The derivatives are ordinary since partial derivatives apply only to functions of many independent variables. Linear differential equations have solutions that can be added and multiplied by coefficients. They are well-defined and understood with exact closed-form solutions. Ordinary differential equations that lack additive solutions are nonlinear. Therefore solving them is more intricate. Graphical and numerical methods applied may approximate solutions of ordinary differential equations and will yield useful information, often sufficing in the absence of exact, analytic solutions.

The notation for differentiation varies depending upon the author and upon which notation is more useful for the task at hand. The general definition of an ordinary differential equation lets F by a given function of x, y and derivatives of y. The equation is as followed

F (x, y, y’,…..y^(n-1)) = y^n

This is called an explicit ordinary differential equation of order n

More generally, an implicit ordinary differential equation of order n takes the form of

F (x, y, y’, y’’,……y^n) = 0

A differential equation not depending on x is called an autonomous. A differential equation is said to be linear if F can be written as a linear combination of the derivatives of y.

A number of coupled differential equations form a system of equations. If y is a vector whose elements are functions; y(x) = [y1(x), y2(x),….,ym(x)]’ and F is a vector valued function of y and its derivatives. 

Numerical Solution of a Second Order BVP

1. Solve the governing equation dealing with the heat transfer phenomena in a heated rod using the Finite Difference method. 2. The heat transfer problems dealing with the radiation phenomena face nonlinear boundary conditions. Use the shooting method to solve the governing equation. 3. The governing differential equation

Solving Second Order Linear Differential Equation

Find the transient and steady-state currents in the RLC circuit with L= 5 henrys, R= 10 ohms, C=0.1 farad and E=25sint volts. L(d^2Q/dt^2) + R(dQ/dt) + 1/C*Q = E refer to attachment for better formula representation.

Runge-Kutta Method Problem

1. If air resistance is proportional to the square of the instantaneous velocity, then the velocity v of a mass m dropped from a given height is determined from: m*dv/dt = mg - kv^, k>0. Let: v(0) = 0, k = 0.125, m = 5 slugs, and g = 32 ft/s^2 a) Use a fourth-order Runge-kutta (RK4) method with h = 1 to approximate the v

Numerical solutions Euler & Runge Kutta

Need to solve and compare the results of the linear vs non linear pendulum problem. Compare the solutions for the approximation (linear) and numerical(non-linear) using the numerical method runge kutta of 4th DEGREE OR HIGHER(preferrably 4th-6th order). Please include the following details: 1. Detailed explanation of methods

Power Series Expansion

a) Give the first few terms in the power series expansion (up to the fourth power) of the solution of the initial value problem: y' = e^x + x cos y , y(0) =0. b) Determine first terms in the power series expansion y =[summation ]a(k)x^k for the solution to the initial value problem : y' = x^3 + y^3 , y(0) = 1. Det

Solving two independent ordinary differential equations

Two tanks A and B, each of volume V, are filled with water at time t=0. For t > 0, volume v of solution containing mass m of solute flows into tank A per second; mixture flows from tank A to tank B at the same rate and mixture flows away from tank B at the same rate. The differential equations used to model this system are giv

Systems of Ordinary Differential Equations

Solve the matrix differential equation X^'=AX where X= [x_1,〖 x〗_2 ]^T=[■(x_1@x_2 )] and A=[■(3&-1@-5&-1)]. Find the eigenvalue(s) of A by solving |λ-A|=0 Solve the linear equation (λ-A)u=0 to get the eigenvector(s) u= 〖[u_1,u_2]〗^2 Find the fundamental matrix Φ(t) What is the Wronskian for Φ? Use

Ordinary Differential Equations

Hi there, I have a question which can be located here can someone please take a look? Full working step by step solution in pdf or word please. If you think the bid is insufficient and you can do it, please respond with a counter bid. Thank you.

Ordinary Differential Equations

Hi there, I have a question regarding ODE which can be located here can someone please take a look? Full working step by step solution in pdf or word please. If you think the bid is insufficient and you can do it, please respond with a counter bid. Thank you.

Ordinary Differential Equations

Hi there, I have a question regarding ODE which can be located here can someone please take a look? Full working step by step solution in pdf or word please. If you think the bid is insufficient and you can do it, please respond with a counter bid. Thank you.

Existence and Uniqueness of Solution to an ODE

See the attached file. Express the 2nd order ODE d_t^2 u=(d^2 u)/(dt)^2 =sin?(u)+cos?(Ï?t) Ï? Z/{0} u(0)=a d_t u(0)=b as a system of 1st order ODEs and verify that there exists a global solution by invoking the global existence and uniqueness Theorem. Useful information: Global existence and uniqueness Theore

Mathematics - Ordinary Differential Equations .

A ball weighing 6lb is thrown vertically downward toward the earth from a height of 1000ft with an initial velocity of 6ft/s. As it falls it is acted upon by air resistances that is numerically equal to 2/3v (in pounds), where v is the velocity (in feet per second) a) What is the velocity and distance at the end of one minute

Mathematics - Ordinary Differential Equations

112. Decreasing cube: Each of the three dimensions of a cube with sides of length s centimeters is decreased by a Whole number of centimeters. The new volume in cubic Centimeters is given by V(s)= s^3-13s^2+54s-72 a) Find V(10). b) If the new width is s - 6 centimeters, then what are the new length and height? c) Find

Linearly independent solutions

Determine another linearly independent solution to y''+2xy'+2y=0 given that one homogeneous solution to this ODE is y_1(x)=exp(-x^2). Verify linear independence by calculating the wronskian.

Linearizing Lorenz Equations using the Implicit Euler Method

I need help to linearize the Lorenz equations so that I can use Matlab to create the butterfly effect, etc. We were given the linearized equations but a couple of students pointed out that one of them was wrong. I don't know which equation is wrong, so if someone could show me how to at least linearize the first Lorenz equatio

Using Substitution to Simplify an ODE

Let a, B be constants. Consider the ODE y'' + (a/x)y' + (B/x^2)y = 0 on the half-line x belongs to (0,infinity). (a) Make the substitution t = ln(x) and write the ODE with independent variable t. (b) From what you know about constant coefficient ODEs, separate the problem into three cases according to the values of a and B

Solving Ordinary Differential Equation using Laplace Transform

(1) Use Laplace Transforms to solve Differential Equation y'' - 8y' + 20 y = t (e^t) , given that y(0) = 0 , y'(0) = 0 (2) Use Laplace Transforms to solve Differential Equation y''' + 2y'' - y' - 2y = Sin 3t , given that y(0)=0 , y'(0)=0 ,y''(0)=0, y'''(0)=1 Note: To see the questions in their mathematic

Differential equation solved with variables separable method

Let x: [0, infinity) -> R and y: [0, infinity) -> R be solutions to the system of differential equations: x' = - x y' = - sin y With initial condition: x(0) = y(0) = alpha, where alpha belongs to [0, pi) (a) Show that |x(t)| =< alpha for all t >= 0 (b) Show that | y(t) -

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(

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

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

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

Matlab Program to Model a Situation

Suppose that a rabbit is initially at point (0,100) and a fox is at (0,0). Suppose that the rabbit runs to the right at speed Vr = 5 ft/sec and the fox always runs toward the rabbit at speed Vf = 6 ft/sec. Write a Matlab program that determines to within 1 second, when the fox catches the rabbit. The program should also plot rab

Sketch direction fields for the following ODE's

Sketch the direction fields for the following ODE's. Make use of isoclines wherever possible. a. y' = y - x + 1 b. y' = 2x c. y' = y - 1 d. y' = xsquared + ysquared - 1 e. y' = y - xsquared Please note y'=y prime. It looks diff, when i see the ? #2. In each direction field above sketch integral curves for which

Using Euler's method

Y'= 1/2 - (X)+2x when y(0)=1 Find the exact solution of ___ O/ <<note!!!! I don't know how to put in a zero with a line going across to make a pheee. 1a. Let h=.1 use euler & improved to approximate to get "Phee" of .1, phee of.2, and phee of .3