Reduction Higher Order Equations Systems

Phaser is designed for systems of first-order ordinary differential equations (ODE). Therefore, when faced with a differential equation involving higher-order derivative, it is necessary to convert it to an equivalent system of first-order equations.

Consider, for instance, the second-order differential equation

    x'' = f(t, x, x')

With the initial conditions

    x(t0) = x0  ,     x'(t0) = x'0   .

To convert this initial-value problem to an equivalent one for a pair of first-order differential equations, which introduce the variables?

    x1 = x
    x2 = x'   .

Now, the differential equations for x1' and x2' become the following pair

    X1’ = x2
    x2' = f(t, x1, x2)

With the initial conditions

    x1 (t0) = x0   ,     x2(t0) = x'0   .

Example: The motions of many simple mechanical systems are governed by Newton's Second Law F = m a, where F is the net force acting on the system, m is mass, a is acceleration which is the second derivative of the displacement with respect to time. a = F/m is a natural second-order differential equation.

As a specific example, let us consider the planar pendulum. In case if we let O be the displacement angle of the pendulum from the vertical position, according to Newton's Second Law, motion of the pendulum is governed by the second-order differential equation

O'' = - (g / l) sin (O) - (c / (l m)) O'   ,

Where g is the gravitational constant, l is the length of the pendulum, m is mass of the pendulum, c is friction constant.

To convert this second-order differential equation to an equivalent pair of first-order equations, so we introduce the variables

    x1 = O
    x2 = O'  ,

that is, x1 is the angular displacement and x2 is the angular velocity. The equations for x1' and x2' become the following pair

    x1' = x2
    x2' = - (g / l) sin(x1) - (c /(l m)) x2.

These are equations stored in ODE Library of Phaser under the name Pendulum ODE. services provide you with a dexterous team of Reduction of Higher Order Equations to Systems assignment writers and assignment helpers and with the unique features it provides will consequently upgrade your results and we assure you of that. Our website provide unique features like

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