1636715336 differential equations handbook

DIFFERENTIAL EQUATIONS

HANDBOOK

I. FIRST ORDER ODEs

i. Separable Variables

a. Standard Form

b. Assumptions

c. Procedure

ii. Linear Equations

a. Standard Form

b. Assumptions

c. Procedure

iii. Exact Equations

a. Standard Form

b. Assumptions

c. Procedure

iv. Solutions by Substitutions

a. Homogeneous Coefficients

1. Standard Form

2. Assumptions

3. Procedure

b. Bernoullis Equations

1. Standard Form

2. Assumptions

3. Procedure

c. Reduction to Separable Equations

1. Standard Form

2. Assumptions

3. Procedure

II. SECOND/HIGHER ORDER ODEs

i. Homogeneous Linear/Constant Coefficients

a. Standard Form

b. Assumptions

c. Procedure

ii. Cauchy-Euler Equations

a. Standard Form

b. Assumptions

c. Procedure

iii. Method of Undetermined Coefficients

a. Superposition Approach

1. Standard Form

2. Assumptions

3. Procedure

b. Annihilator Approach

1. Standard Form

2. Assumptions

3. Procedure

c. Reduction of Order

1. Standard Form

2. Assumptions

3. Procedure

iv. Variation of Parameters

a. Standard Form

b. Assumptions

c. Procedure

v. Laplace Transforms

a. Standard Form

b. Assumptions

c. Procedure

vi. Series Solutions

a. Method for Ordinary Point Solutions

1. Standard Form(s)

2. Assumptions

3. Procedure

b. Method for Singular Point Solutions

1. Standard Form(s)

2. Assumptions

3. Procedure

III. LINEAR SYSTEMS

i. First Order

a. Undetermined Coefficients

1. Standard Form

2. Assumptions

3. Procedure

b. Variation of Parameters

1. Standard Form

2. Assumptions

3. Procedure

c. Matrix Exponential Method

1. Standard Form

2. Assumptions

3. Procedure

d. Laplace Method

1. Standard Form

2. Assumptions

3. Procedure

ii. Second Order to Autonomous System

a. Standard Form

b. Assumptions

c. Procedure

IV. VARIATIONAL FORMULATIONS

i. Differential Form D Form

a. Standard Form

b. Assumptions

c. Procedure

ii. Variational Form V Form

a. Standard Form

b. Assumptions

c. Procedure

iii. Minimizational Form M Form

a. Standard Form

b. Assumptions

c. Procedure

V. APPLICATION OF NUMERICAL METHODS

i. Eulers Method (First Order)

a. Standard Form

b. Assumptions

c. Procedure

ii. Runge-Kutta

a. Standard Form

b. Assumptions

c. Procedure

iii. Weighted Residuals Methods

a. Least Squares Method

b. Collocation Method

c. Galerkins Method

d. Subdomain Method

VI. PDEs i. Integral Transforms

a. Laplace Transforms

1. Standard Form

2. Assumptions

3. Procedure

b. Fourier Transforms

1. Standard Form

2. Assumptions

3. Procedure

I. FIRST ORDER ODEs

i. Separable Variables

a. Standard Form

( ) ( ) ( )

b. Assumptions

( ) can be separated into a product of the functions ( ) and

( ).

c. Procedure

1. Separate the variables in x and y to obtain and integrate.

( ) ( )

ii. Linear Equations (First Order)

a. Standard Form

( ) ( )

b. Assumptions

( ) can be separated into a product of the functions ( ) and

( ). This procedure is also known as Variation of Parameters.

c. Procedure

1. Find the homogeneous solution to this DE. The DE is now

separable. Obtain solution :

( )

2. Find homogeneous solution, :

( )

3. Find integrating factor, :

( )

4. Compute particular solution, :

5. General solution is:

iii. Exact Equations

a. Standard Form

( ) ( )

b. Assumptions

( )

( )

c. Procedure

1. Let:

( )

2. Obtain ( ) by integrating:

( ) ( ) ( )

3. Differentiate constant of integration, ( ), with respect to :

( )

4. Plug in ( ) and match with ( ):

( )

( ) ( ) ( )

5. As an alternative, you can start with:

( )

and integrate/differentiate as appropriate.

iv. Solutions by Substitutions

a. Homogeneous Coefficients

1. Standard Form

( ) ( )

2. Assumptions

All terms within the coefficient functions, and , are of the same

degree:

( ) ( ) ( ) ( )

3. Procedure

(i) Let:

(ii) Substitute to obtain a separable equation using:

or

(iii) Arrange terms in separable equation form and integrate.

b. Bernoullis Equations

1. Standard Form

( ) ( )

2. Assumptions

is any real number.

3. Procedure

(i) Let:

(ii) Substitute for and

to obtain a linear DE in using:

(iii) The DE is now a linear equation in .

c. Reduction to Separable Equations

1. Standard Form

( )

2. Assumptions

3. Procedure

(i) Let:

(ii) Substitute for ( ) and

to obtain a separable equation

in using:

(iii) Solve as a separable DE in and obtain a solution, ( ).

(iv) Back substitute for .

II. SECOND/HIGHER ORDER ODEs

i. Homogeneous Linear w/ Constant Coefficients

a. Standard Form

( )

( )

( )

b. Assumptions

( ) ( )

( ) * + ( )

c. Procedure

1. Solve the order auxiliary equation for roots, :

2. Acquire the general solution from the roots:

Case

(i) For Distinct Real Roots

(ii) For , Repeated Real Roots

(iii) For Conjugate Complex Roots,

3. Obtain from Method(s) II.iii or II.iv.

ii. Cauchy-Euler Equations

a. Standard Form

( )

( )

( )

b. Assumptions

( )

( )

Procedure


2 Acquire from the roots:

Case

(i) For Distinct Real Roots

(ii) For , Repeated Real Roots

( )

(iii) For Conjugate Complex Roots,

, ( ) ( )-

3. Obtain from Method(s) II.iii or II.iv.

iii. Method of Undetermined Coefficients

a. Superposition Approach

1. Standard Form

( )

( )

( )

2. Assumptions

( )

* + ( )

3. Procedure

(i) Find the complementary solution, , by solving the

homogeneous DE (set ( ) ). See Method II.i.

(ii) Assume a particular solution, , that is similar in form to

the forcing function, ( )

Case

(1) For ( ) is an exponential function,

(2) For ( ) is a trigonometric function, ( ) or ( ).

( ) ( )

(3) For ( ) is a polynomial function,

(4) For ( ) is a product of 2 or more of the above functions.

The product of the particular solutions for all individual

factors, will be, .

( ) ( )

(iii) SUPERPOSITION PRINCIPLE - For ( ) is a linear

combination of 2 or more of the above functions. The sum

of the particular solutions for the individual terms, will be,

.

( ) ( ) ( )

(iv) When terms for and are similar, multiply the particular

solution by , where r is the smallest positive integer which

will eliminate duplication. Consider the DE:

(v) Substitute into the original DE and formulate a system of

undetermined coefficient equations by matching like terms

with the forcing function of the DE. Use matrices if

necessary to solve for undetermined coefficients.

(vi) The general solution will be:

Trial Particular Solutions

Function, ( ) Form of 1

( ) ( )

( )

( ) ( )

( ) ( )

b. Annihilator Approach

1. Standard Form

( )

( )

( )

2. Assumptions

( )

* + ( )

3. Procedure

(i) Find the complementary solution, , by solving the

homogeneous DE (set ( ) ). See Method i.

(ii) Find the differential operator, , which will annihilate the

forcing function, ( ).

Case

(1) The differential operator for the following exponential

functions is, ( ) :

, , , ,

(2) The differential operator for the following trigonometric

functions is, , ( )- :

, , , ,

, , , ,

(3) The differential operator for the following polynomial

terms is, :

, , , ,

(iii) Express the DE in terms of the differential operator, .

(iv) Combine the differential operators for the forcing function

and DE to obtain:

( ) DE in differential operator form

Annihilator for forcing function

Auxiliary equation in terms of

(v) Solve the auxiliary equation for roots, , and determine

and from Method i Step 3 (Homogeneous cases).

(vi) Substitute into the original DE and formulate a system of

undetermined coefficient equations by matching like terms

with the forcing function of the DE. Use matrices if

necessary to solve for undetermined coefficients.

c. Reduction of Order

1. Standard Form

( ) ( ) ( )

( )

2. Assumptions

( )

( ) ( ) * +

3. Procedure

(i) If ( ) , find the second solution, , by using the

following formula:

( ) ( )

( )

(ii) The general solution is:

(iii) If ( ) , first find the solution to , then find , using

Method(s) iii or iv.

iv. Variation of Parameters

a. Standard Form

( ) ( )

( )

( )

b. Assumptions

( )

( ) ( ) * +

c. Procedure


2. Acquire from the roots. Use Methods ii or iii to

determine the complementary functions:

3. The particular solution is a linear combination of the

product of a variable parameter ( ) and each

complementary function ( ):

( ) ( ) ( ) ( ) ( ) ( )

4. Find the Wronskian for the homogeneous DE.

[

( )

( )

]

5. Set up an column matrix for the forcing function..

, ( )-

6. Find the Wronskian, , for each , by replacing the

column in with the column created in the previous step.

[

( ) ( )

]

7. With Wronskians, we can now solve for the variable

parameters, .

( )

8. The general solution is:

v. Laplace Transforms

a. Standard Form

( )( )

( )

( )

b. Assumptions

, ( )

( ) ( )( ) ( ) ( )

( )

( ) * +

( )

* +

c. Procedure

1. Take the Laplace transform of each of the terms in the DE.

[ ( )]

, ( )-

2. Combining like terms, substitute in zero-state conditions to

form an equation in , as such:

( ) ( ) ( ) ( )

3. Represent each term in as a sum of rational functions

whose Laplace transforms are known, ie.:

( )

( )( )

( )

( )

4. Take the Inverse Laplace of each term to obtain the

solution, ( ).

( ) , ( )- , ( )- , ( )- , ( )-

( )

, ( ) ( )-

Inverse Laplace Transform Table

Term, ( ) Inverse Laplace, ( ) * ( )+ ( ) * ( ) +

( )

( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) (

)

( )

( )

( )

( )

( )

( )

( )

( )

( )

( )

( )

( )

( )

( )

( )

( )

( )

( )

( )

( )

( )

( )

( )

( ) ( )

( )

( ) ( )

Laplace Transform Table

Function, ( ) Laplace Transform, ( ) * ( )+ ( )

( )

1

( )

( )

( )

( )

( )( ) ( ) ( ) ( ) ( )( )

( ) ( )

( ) ( ) ( )

( ) ( ) * ( )+

( ) ( )

( )

( ) ( )

( ) ( )

( ) ( )

( )

( )

( )

( )

( )

( )

vi. Series Solutions

a. Method for Ordinary Point Solutions

1. Standard Form

( ) ( )

( )

( ) ( ) ( ) ( )

( ) ( ) ( )

( )

2. Assumptions

( ) ( )

( ) * +

( )

3. Procedure

(i) Observe the point at which a solution is desired and

determine whether or not there are singular points for the

DE. Adjust the interval of convergence, accordingly.

(ii) If the solution desired is not at an ordinary point go to

Method b, otherwise assume solutions of the form:

( )

The general solution will be of the form:

( )

( )

where are the coefficients of the general solution.

(iii) Express each term in the DE as a series incorporating the

factors within it. Starting index will equal order of derivative

taken, ie.:

( )

( )( )

( )( )

(iv) Compare all newly formed series terms and replace each

index with a new dummy index, , which will generate a

common factor of for all series terms, ie.:

( )( )

( )( )

( )

( )( )

( )( )

( ) ( )

( )

(v) Identify the series (Series and ) with the highest starting

index ( ) and shift all other series to that index value

by taking out terms from the summation:

( )( )

( )( )

( )

( )

( )

( )

( )

(vi) Substitute newly found series terms into original DE and

arrange them under one summation sign in order to yield a

recurrence relation:

( )

,( )( ) ( )( )

( ) -

(vii) Formulate the recurrence and indicial equations:

( )( ) ( )( ) ( )

[( )

( )

( ) ] ( )

( )

(viii) Establish initial value sets from the constants found in

( ) by initializing the first constants, ie.:

(ix) Calculate the remaining constants in each set utilizing the

recurrence relation in ( ). Use calculator if necessary.

Set 1

0 1 2 3 4 5 6 7 1 0 -1/2 -1/6 1/8 1/12 -1/42 1/112

Set 2

0 1 2 3 4 5 6 7 0 1 1 0 -1/4 1/20 1/40 -1/56

(x) Arrange the series solutions to the DE:

( ) (

)

( ) (

)

(xi) Apply any initial conditions (ie. ( ) ( ) ) to

these equations to obtain the coefficients of the general

solution , and .

(xii) The general solution is:

( ) (

)

(

)

b. Method for Singular Point Solutions

1. Standard Form

( ) ( )

( )

( ) ( ) ( ) ( )

( ) ( ) ( )

( )

2. Assumptions

( ) ( )

( ) * +

( )

If is a singular point, the following statement is true:

( ) ( ) * , - +

3. Procedure

(i) Observe the point at which a solution is desired and identify

the irregular/regular singular points within the DE. This

solution method is valid for a DE with regular singular

points only. Adjust the interval of convergence,

accordingly.

(ii) Assume solutions of the form:

( )

( )

The general solution will be of the form:

( )

( )

where are the coefficients of the general solution.

(iii) Express each term in the DE as a series incorporating any

factors within it while maintaining the starting index at

.:

( )

( )( )

( )( )

(iv) Compare all newly formed series terms and replace each

index with a new dummy index, , which will generate a

common factor of for all series terms, ie.:

( )

( )( )

( )( )

( )

( )

( )

( )

( )

( )

( )

(v) Identify the series (Series and ) with the highest starting

index ( ) and shift all other series to that index value

by taking out terms from the summation:

( )( )

( )( )

( ) ( )( )

( ) ( )

( ) ( )

(vi) Substitute the newly found series terms into the original DE

and arrange them under one summation sign in order to

yield a recurrence relation and an indicial equation:

[* ( )( ) ( ) + ( )

]

( )( ) ( )

, ( )( ) ( ) ( ) -

(vii) Obtain roots for the indicial equation:

(viii) Substitute each root , into the recurrence relation and

obtain recurrence equations.

( )( ) ( )

[

( )] [

] ( )

(

) (

) (

) (

)

[

( )] [

] ( )

(ix) Evaluate a set of in terms of for each recurrence eqn.:

(x) Calculate the remaining constants in each set utilizing a

calculator if necessary.

Set 1

0 1 2 3 4 5 6 7 1 1/2 1/10 1/80 1/880 1/12,320 --- ---

Set 2

0 1 2 3 4 5 6 7 1 1/3 1/18 1/162 1/1944 1/29,160 --- ---

(xi) Arrange the series solutions to the DE:

( ) (

)

( ) (

)

(xii) Apply any initial conditions (ie. ( ) ( )

) to

these equations to obtain the coefficients of the general

solution, and .

(xiii) The general solution is:

( ) (

)

(

)

III. LINEAR SYSTEMS

i. First Order a. Undetermined Coefficients

1. Standard Form

( ) ( )

( )

2. Assumptions

( ) ( ) * +

3. Procedure

(i) Create a homogeneous linear system by letting

( ) .

(ii) Formulate matrices from standard form.

(

)

( ) ( ( ) ( )

( ) ( )

), (

)

(iii) Obtain the eigenvalues for the matrix.

(iv) Generate a complementary solution for the

homogeneous case according to the following:

Case

(1) For Distinct Real Roots

(2) For Repeated Real Roots of multiplicity,

(a) When a repeated eigenvalue generates distinct

eigenvectors:

(b) When a repeated eigenvalue generates only one

eigenvector use the following method in

succession to generate distinct eigenvectors:

( )

( )

( )

( )

The homogeneous solution will then be:

[

]

, -

(3) For Conjugate Complex Roots,

, ( ) ( )-

, ( ) ( )-

( ) ( )

(v) To obtain the particular solution, , for the non-

homogeneous case, assume a form for the particular

solution:

Trial Particular Solutions

Function, ( ) Form of

(

) (

)

(

) (

) (

)

(

) (

) (

) (

)

(

) (

) (

) (

) (

)

(

) (

) (

)

(

) (

)

(

) (

) (

) (

) (

)

(vi) Substitute into the original DE and formulate a

system of undetermined coefficient equations by

matching like terms with the forcing function of the DE.

Use matrices if necessary to solve for undetermined

coefficients.

(vii) The general solution will be:

b. Variation of Parameters

1. Standard Form

( ) ( )

( )

2. Assumptions

( ) ( ) * +

3. Procedure

(i) Follow steps 1 through 4 for Method IV.i.

(ii) Create the fundamental matrix, ( ), from the solution

vectors found in previous steps:

( ) (

)

(iii) Calculate the inverse of the fundamental matrix, ( ).

(iv) Calculate the particular solution from the following

formula:

( ) ( ) ( )

(v) The general solution is:

c. Matrix Exponential Method

1. Standard Form

( ) ( )

( )

2. Assumptions

( ) ( ) * +

3. Procedure

(i) Identify the A matrix.

(ii) Construct the fundamental matrix,(t), from the A matrix using the following identity substituting any known power

series identities if necessary (ie. ).

( )

(iii) For the Initial Value Problem ( ) only, we can use the

inverse Laplace Transform of .

( ) *( ) +

(iv) Calculate the inverse of the fundamental matrix, ( )

replacing t by s.

( ) ( )

(v) Calculate the general solution.

( ) ( ) ( ) ( )

( ) , ( )-

d. Laplace Method

1. Standard Form

( )

( )

( )

2. Assumptions

( ) ( ) * +

* +

3. Procedure

(i) Take the Laplace Transform of each of the terms in the DEs

to obtain a linear system in terms of .

[

] [ ( ) ]

, ( )-

(ii) Formulate matrices from the previous step.

( ) ( ( )

( )

)

( ) ( ( ) ( )

( ) ( )

), ( ) ( ( )

( )

)

(iii) Solve for the matrix using the following equation:

(iv) Take the Laplace Transform of each of the terms in .

ii. Second Order to Autonomous System

a. Standard Form

( )

( )

( )

( )

b. Assumptions

( ) * +

c. Procedure

1. Reduce the order of the order DE by linearizing the

dependent variable and its derivatives.

( )

2. Formulate the matrices and ( ).

IV. VARIATIONAL FORMULATIONS

i. Differential Form D Form

a. Standard Form

( )

( )

b. Assumptions

( )

( ) ( )

c. Procedure

1. Linearize the problem by formulating the equation of the

tangent line at the given point ( ).

( ) ( )( )

2. Create successive tangent lines from the initial point to

the point to be approximated ( ) at a distance from

one another using the following recursive eqn.:

( )

3. Stop when . Decrease the step size to arrive at a

closer approximation to .

ii. Variational Form V Form

a. Standard Form

( )

( )

b. Assumptions

( )

( ) ( )

c. Procedure



( ) ( )( )




( )



iii. Minimizational Form M Form

a. Standard Form

( )

( )

b. Assumptions

( )

( ) ( )

c. Procedure



( ) ( )( )




( )



V. APPLICATIONS OF NUMERICAL

METHODS

i. Eulers Method (First Order)

a. Standard Form

( )

( )

b. Assumptions

( )

( ) ( )

c. Procedure



( ) ( )( )




( )



1636715336 differential equations handbook

Documents