Differential Equations - First Order Ordinary Differential Equations

Jumat, 07 Agustus 2020

Functions of one variable: y, p, q, u, g, h, G, H, r, z

Arguments (independent variables): x, y

Functions oftwo variables: f(x, y), M(x, y), N(x, y)

First order derivative: y’, u’,, …

Partial derivative:

Natural number: n

Particular solutions: y₁, y_p

Real numbers: k, t, C, C₁, C₂, p, q, α, β

Roots of the characterististic equations: λ₁, λ₂

Time: t

Temperature: T, S

Population function: P(t)

Mass of an object: m

Stiffness of a spring: k

Displacement of the mass from equilibrium: y

Amplitude of the displacement: A

Frequency: ω

Damping coefficient: γ

Phase angle of the displacement: δ

Angular displacement: θ

Pendulum length: L

Acceleration of gravity: g

Current: I

Resistance: R

Inductance: L

Capacitance: C

 

First Order Ordinary Differential Equations

 

• Linear Equations

     .

   The general solution is

,

    where

.

 

• Separable Equations

     

    The general solution is given by

      ,

     or

     H(y) = G(y) + C.

 

• Homogeneous Equations

    The differential equation

     is homogeneous, if the function f(x, y) is homogeneous, 

         that is f(tx, ty) = f(x, y). 

   The substitution(then y = zx) leads to the 

   separable equation.

 

• Bernoulli Equations

      .

    The substitution z = y ^{1 – n} leads to the linear equation

      .

 

• Riccati Equations

     

    If a particular solution y₁ is known, then the general solution can be 

    obtained with the help of substitution, which leads to the 

    first order linear equation

      .

 

• Exact and Nonexact Equations

    The equation

     M(x, y) dx + N(x, y) dy = 0

     is called exact if, and nonexact otherwise.

 

     The general solution is

       

 

• Radioactive Decay

     ,

    where y(t) is the amount of radioactive element at time t, k is the rate 

     of decay.

    The solution is y(t) = y₀ e ^{– ky} , where y₀ = y(0) is the initial amount.

 

• Newtor’s Law of Cooling

     ,

   where T(t) is the temperature of an object at time t, S is the temperature 

    of the surrounding environment, k is a positive constant.

 

    The solution is

    T(t) = S + (T₀ – S) e ^{– kt},

    where T₀ = T(0) is the initial temperature of the object at time t = 0.

 

• Population Dynamics (Logistic Model)

     ,

   where P(t) is population at time t, k is a positive constant, M is 

    limiting size for the population.

 

   The solution of the differential equation is, 

   where P₀ = P(0) is the initial population at time t = 0.

 

 

Sumber

Labels: Mathematician

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