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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.

### Ordinary Differential Equations - The Laplace Transform

a) Use the definition of the Laplace transform b) Use the properties of the Gamma function to calculate.. c)Use the Laplace properties and the s-shifting theorem to solve the differential equation see attached

### 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.

### Solving Second Order Linear Differential Equation by Using D-operator

Use D-operators to find a particular solution to the differential equation: y^n + y' - 2y= e^-2x Hence write its general solution. Find the solution that satisfies the initial conditions: y(0) = 1/3, y'(0) = -1/3

### 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

### Lipschitz continuity and its role in the existence and uniqueness of ordinary differential equations is investigated.

PROBLEM 1. Find a Lipschitz constant, K, for the function f (u, t) = u^3 + t u^2 which shows that f is Lipschitz in u on the set 0 ? u ? 2, 0 ? t ? 1. PROBLEM 2. Show that the function f (u, t) = t u^(1/2), is not Lipschitz in u on [0, 1] × [0, 2]. PROBLEM 3. Find two solutions to the initial value problem y = |y|^(1/2) ,

### 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.

### Improved Euler method with Butcher Tableau

Use the order conditions to prove that the improved Euler method with Butcher Tableau 0 | 1 | 1 ---+------------- | 1/2 1/2 is second order but not third order.

### 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

### Ordinary differential equation: Example problem

Solve the ode y'' - y' +4y = 0 as a system of first order odes.

### Finding the Solutions to Ordinary Differential Equations

Please help solve #2. Find the solution of the ODE: y'' + y = cot2t Please see the attached file for the fully formatted problem.

### 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

### Euler's Method

Use Eulers method with step size 0.1 to estimate y(0.5), where y(x) is the solution of the initial-value problem y ' = 3y + 3xy, y(0) = 1. (Give the answer to four decimal places.) y(0.5) =

### Particular solution of a linear ordinary differential equation.

Find the particular solution of a linear ODE subject to the given initial condition: (Attached). y' + 3/x = x y(1) = 0

### Example of Finding a Solution for a Differential Equation

Find the particular solution of the following differential equation: 12(d^2y/dt^2)-3y=0 given that when t=0, y=3 and dy/dt=0.5 and could you explain the reasons for choosing y=e^(rt).

### Differenial Equations : Fundamental Solutions and a Reflectionless Potential

Please see the attached file for the fully formatted problems.

### Solving Ordinary Differential Equations

Explain how to solve differential equations of the following types: y'=f(ax+by+c) y'=f(y/x) x=f(y,y') Provide examples with complete solutions for each case.

### 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

### Second Order Differential Equation - RHS Equal to Zero

Solve a second order differential equation with RHS equal to zero. See attached file for full problem description.