Differential Equations - Second Order Ordinary Differential Equation - Some Partial Differential Equations

Sabtu, 08 Agustus 2020

•       Homogeneous Linear Equations with Constant Coefficient

y” + py’ + qy = 0.

The charanteristic equation is

λ² + pλ + q = 0.

If λ₁ and λ₂ are distinct real roots of the characteristic equation, then the general solution is

, where

C₁ and C₂ are integration constants.

  

If, then the general solution is.

If λ₁ and λ₂ are complex numbers:

λ₁ = α + βi, λ₂ = α – βi where

,

Then the general solution is

y = e ^αx (C₁ cos βx + C₂ sin βx).

 

•        Inhomogeneous Linear Equations with Constant Coefficient

y” + py’ + qy = f(x).

 

The general solution is given by

y = y_p + y_h, where

y_p is a particular solution of the inhomogeneous equation and y_h is the general solution of the associated homogeneous equation.

 

If the right side has the form

f (x) = e ^αx (P₁(x) cos βx + P₁(x) sin βx),

then the particular solution y_p is given by

y = x^k e ^αx (R₁ cos βx + R₂ sin βx),

where the particular R₁ (x) and R₂ (x) have to be found by using the method of undetermined coefficients.

• If α + βi is not a root of the characteristic equation, then the power k = 0,

• If α + βi is a simple root, then k = 1,

• If α + βi is a double root, then k = 2.

 

•        Differential Equation with y Missing

y" = f (x, y’).

Set u = y’. then the new equation satisfied by v is u’ = f (x, u),

Which is a first order differential equation.

 

•        Differential Equation with x Missing

y" = f (y, y’).

Set u = y’. Since

,

we have

,

Which is a first order differential equation.

 

•        Free Undamped Vibrations

The motion of a Mass on a Spring is described by the equation

,

Where

m is the stiffness of the spring,

y is displacement of the mass from equilibrium.

 

The general solution is

y = A cos (ω₀ t – δ),

where

A is the amplitude of the displacemet,

ω₀ is the fundamental frequency, the period is,

δ is phase angle of the displacement.

This is an example of simple harmonic motion.

 

•        Free Damped Vibrations

  , where

γ is the damping coefficient.

There are 3 cases for the general solution:

Case 1. γ² > 4km (overdamped)

 

Case 2. γ² = 4km (critically damped)

 

 

Case 3. γ² < 4km (underdamped)

 

•        Simple Pendulum

   ,

Where θ is the angular displacement, L is the pendulum length, g is the acceleration of gravity.

 

•        RLC Circuit

    ,

Where I is the current in an RLC circuit with an ac voltage source V(t) = E₀ sin (ωt).

 

The general solution is

C₁, C₂ are constants depending on initial conditions.

 

 

Some Partial Differential Equations

•        The Laplace Equation

    

Applies to potential energy function u(x, y) in the xy-plane where heat is allowed to ftow from warm areas to cool ones. The equations of this type are called parabolic.

 

•        The Heat Equation

    

Applies to the temperature distribution u(x, y) for a conservative force field in the xy-plane. Partial differential equations of this type are called elliptic.

 

•        The Wave Equation

   

Applies to the displanement u(x, y) of vibrating membranes and other wave function. The equations of this type are called hyperbolic.

 

Sumber

Labels: Mathematician

Thanks for reading Differential Equations - Second Order Ordinary Differential Equation - Some Partial Differential Equations . Please share...!