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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 http://nullspace8.blogspot.com/2011/10/e2_26.html. 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 http://nullspace8.blogspot.com/2011/10/e3.html. 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 http://nullspace8.blogspot.com/2011/10/e2.html. 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) -

Series Solutions of Second Order Linear Equations

Verify that x0=0 is an ordinary point of the differential equation: y''+ xy' + 2y = 0 Find two linearly independent solutions to the differential equation in the form of a power series about x0=0. If possible, find the general term in each solution. Write the general solution Verify that x0=0 is an ordinary point for the

Solving an Ordinary Differential Equation

DP/dt = m(a0)[exp(-z1)t] - (z2/z1)P Solve this differential equation with a = ao at t=0 and a=a at t=t to show that: P = [mz1(a)] / [z2 - z12] + [mz1(ao) / (z12- z2)](a/ao)^(z2/z12) Where the last term in this equation is a/ao "raised to the power of" z2/z12 ---