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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. 

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

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

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

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

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

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

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