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Differential Equations

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http://tutorial.math.lamar.edu/terms.aspx

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Introduction to �Differential Equations (DE)

  • Recall basic definitions of DE,
    • order
    • linearity
    • initial conditions
    • solution
  • Classify DE based on( order, linearity, conditions)
  • Classify the solution methods

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Derivatives

Derivatives

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Partial Derivatives

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u is a function of

more than one

independent variable

Ordinary Derivatives

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y is a function of one

independent variable

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Differential Equations

Differential

Equations

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involve one or more

partial derivatives of

unknown functions

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Ordinary Differential Equations

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involve one or more

Ordinary derivatives of

unknown functions

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Partial Differential Equations

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Differential Equations

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Differential Equations (DE) involve one or more ordinary derivatives of unknown functions with respect to one independent variable

y(x): unknown function

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x: independent variable

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differential equation

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The order of an differential equations is the order of the highest order derivative

Second order ODE

First order ODE

Second order ODE

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Solution of a differential equation

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A solution to a differential equation is a function that satisfies the equation.

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Linear DE

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An DE is linear if the unknown function and its derivatives appear to power one. No product of the unknown function and/or its derivatives

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Linear ODE

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Linear ODE

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Non-linear ODE

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Boundary-Value and Initial value Problems

Boundary-Value Problems

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  • The auxiliary conditions are not at one point of the independent variable
  • More difficult to solve than initial value problem

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Initial-Value Problems

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  • The auxiliary conditions are at one point of the independent variable

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same

different

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Classification of DE

DE can be classified in different ways

  • Order
    • First order DE
    • Second order DE
    • Nth order DE
  • Linearity
    • Linear DE
    • Nonlinear DE
  • Auxiliary conditions
    • Initial value problems
    • Boundary value problems

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Solutions

  • Analytical Solutions to DE are available for linear DE and special classes of nonlinear differential equations.

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  • Numerical method are used to obtain a graph or a table of the unknown function

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  • We focus on solving first order linear DE and second order linear DE and Euler equation

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First Order Linear Differential Equations

  • Def: A first order differential equation is said to be linear if it can be written

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First Order Linear Differential Equations

  • How to solve first-order linear ODE ?

Sol:

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Multiplying both sides by , called an integrating factor, gives

assuming we get

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First Order Linear Differential Equations

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By product rule, (4) becomes

Now, we need to solve from (3)

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First Order Linear Differential Equations

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to get rid of one constant, we can use

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Summary of the Solution Process

  • Put the differential equation in the form (1)
  • Find the integrating factor, using (8)
  • Multiply both sides of (1) by and write the left side of (1) as
  • Integrate both sides
  • Solve for the solution

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Example 1

Sol:

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Example 2

Sol:

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�Second Order Linear Differential Equations

  • Homogeneous Second Order Linear Differential Equations
    • real roots, complex roots and repeated roots

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  • Non-homogeneous Second Order Linear Differential Equations
    • Undetermined Coefficients Method

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  • Euler Equations

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where a, b and c are constant coefficients

Let the dependent variable y be replaced by the sum of the two new variables: y = u + v

Therefore

If v is a particular solution of the original differential equation

The general solution of the linear differential equation will be the sum of a “complementary function” and a “particular solution”.

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Second Order Linear Differential Equations

The general equation can be expressed in the form

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Let the solution assumed to be:

characteristic equation

Real, distinct roots

Double roots

Complex roots

The Complementary Function (solution of the homogeneous equation)

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Real, Distinct Roots to Characteristic Equation

  • Let the roots of the characteristic equation be real, distinct and of values r1 and r2. Therefore, the solutions of the characteristic equation are:
  • The general solution will be
  • Example

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Let

where V is a

function of x

Equal Roots to Characteristic Equation

  • Let the roots of the characteristic equation equal and of value r1 = r2 = r. Therefore, the solution of the characteristic equation is:

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Complex Roots to Characteristic Equation

Let the roots of the characteristic equation be complex in the form r1,2 =λ±µi. Therefore, the solution of the characteristic equation is:

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(I) Solve

characteristic equation

Examples

(II) Solve

characteristic equation

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When g(x) is a polynomial of the form where all the coefficients are constants. The form of a particular solution is

Non-homogeneous Differential Equations (Method of Undetermined Coefficients)

When g(x) is constant, say k, a particular solution of equation is

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Example

Solve

equating coefficients of equal powers of x

characteristic equation

complementary function

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Non-homogeneous Differential Equations (Method of Undetermined Coefficients)

  • When g(x) is of the form Terx, where T and r are constants. The

form of a particular solution is

  • When g(x) is of the form Csinnx + Dcosnx, where C and D are

constants, the form of a particular solution is

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Example

Solve

characteristic equation

complementary function

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Example

Solve

characteristic equation

complementary function

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Example

Solve

characteristic equation

complementary function

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Euler Equations

  • Def: Euler equations

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    • Assuming x>0 and all solutions are of the form y(x) = xr
    • Plug into the differential equation to get the characteristic equation

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Solving Euler Equations: (Case I)

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  • The characteristic equation has two different real solutions r1 and r2.
  • In this case the functions y = xr1 and y = xr2 are both solutions to the original equation. The general solution is:

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Example:

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Solving Euler Equations: (Case II)

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  • The characteristic equation has two equal roots r1 = r2=r.
  • In this case the functions y = xr and y = xr lnx are both solutions to the original equation. The general solution is:

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Example:

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Solving Euler Equations: (Case III)

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  • The characteristic equation has two complex roots r1,2 = λ±µi.

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Example: