Differential Equations: Modeling, Solving and Applications, Slides of Mathematics

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Differential Equations

Lê Xuân Trường

Modeling with Differential Equations

Exponential growth model

N(t) is the population size at time t The rate of growth of the population is proportional to the pop- ulation dN dt

= kN(t) (eq1)

In the initial time we have N( 0 ) = N 0

How we can determine N(t)?

The equation is a first-order differential equation

Modeling with Differential Equations

Height of a Moving Baseball A baseball is thrown upward from a height of 3 meters above Earth’s surface with an initial velocity of 10m/s, and the only force acting on it is gravity. The ball has a mass of 0.15kg at Earth’s surface.

Find the position x (t) of the baseball at time t.

Modeling with Differential Equations

Modelling

The force acting on a baseball is the force of gravity

F = mg

Using Newton’s second law of motion we obtain

mg = mx′′(t) or x′′(t) = g

Cauchy problem for second-order differential equation

x′′(t) = g , x ( 0 ) = 3 , x′( 0 ) = 10.

Separable Equations

Examples

1. Solve the following differential equation

dy dx = 6 y 2 x.

2. Solve the following Cauchy’s problem and find the interval of va- lidity of the solution

y ′^ =

xy 3 √ 1 + x^2

, y ( 0 ) = − 1.

First-order linear differential equation

Definition General form: a(x)y’ + b(x)y = c(x) Standard form: y’ + p(x)y = q(x)

q(x ) = 0: Homogeneous equation q(x ) 6 = 0: Nonhomogeneous equation

First-order linear differential equation

Examples

1. Find a general solution for the differential equation

xy ′^ − 2 y = x^2

2. Solve the initial-value problem { y ′^ − 3 ytan(x ) = 1 y ( 0 ) = 1

Second-order linear differential equation

Definition

General form:

a(x )y ′′^ + b(x )y ′^ + c(x )y = d (x )

Standard form:

y ′′^ + p(x )y ′^ + q(x )y = f (x )

f (x ) = 0: Homogeneous equations f (x ) 6 = 0: Nonhomogeneous equations

In this course we only study the case of constants coefficients

Second-order linear differential equation

Examples Find the general solutions of following equations

y ′′^ + 2 y ′^ − 3 y = 0 y ′′^ + 4 y ′^ + 4 y = 0 y ′′^ + y = 0

Second-order linear differential equation

Solving nonhomogeneous equations

y” + py’ + qy = f(x)

y (x ) = ypar (x ) + y 0 (x )

y 0 is the general solution of the homogeneous equation

y” + py’ + qy = 0

ypar is a particular solution of the nonhomogeneous equa- tions

Định lý [General solutions]

Second-order linear differential equation

How to find the particular solution

Case 2: (where Pn and Qm are polynomial of degree n and m)

f (x ) = [Pn(x ) cos β x + Qm(x ) sin β x (^) ] exp(αx )

ypar (x ) = xλ^

[̂

Ps (x ) cos β x + Q̂ s (x ) sin β x

]

exp(αx )

where

λ =

0 , α ± iβ are not solutions of the characteristic equation 1 , α ± iβ are solutions of the characteristic equation

P̂ s and Q̂ s are polynomial of degree s = max{m, n}

Second-order linear differential equation

Examples Solve the following equations

y ′′^ + 4 y ′^ − 5 y = x exp(x ) y ′′^ − 6 y ′^ + 9 y = cos(x )