Second-order Linear Differential Equation

General Form:

( ) ( ) ( )

If q(x) = 0, the equation is said to be homogenous, otherwise, it is non-homogenous.

Existence and Uniqueness Theorem

Let ( ) ( ) and ( ) be continuous on the interval [a,b], then the differential

equation ( ) ( ) ( ) with initial condition ( ) and ( ) has a

unique solution defined for all x in [a,b].

Test for Linear Independence

WRONSKIAN

- The determinant of an array of coefficients that determines the linear independence of

solutions to a differential equation.

A 2nd order Differential Equation with a general solution and initial

conditions and . The initial condition means that the constant and have to satisfy

these equations:

Using Cramer’s rule:

|

|

| |

|

|

| |

Mathematically, the test for linear independence or commonly known as Wronskian test is

define by the equation,

( ) |

|

( )

Note that and is undefined if the denominator |

| or the Wronskian is equal to

zero. Otherwise, we can conclude that and is linearly independent.

Abel’s Theorem

Given that and are any solutions of the equation ( ) ( ) .

Then, ( ) ∫ ( ) where C depends on and .

Superposition Theorem

For ( ) ( ) and have solutions, ( ) and ( ), then any linear combinations of

these solutions are also a solution of the differential equation.

In short, the theory simply says that if 2nd order Differential Equation has the different solutions, a linear

combination of those two solutions is also a solution of the D.E. This theorem is applicable to all forms of

2nd order linear Differential Equation.

Homogeneous Second-order Linear Differential Equation with constant coefficients

where: a, b, and c are constants.

Solution:

1. Substitute,

and

2. The equation simplifies to which is a quadratic equation, this is known

as characteristic equation.

3. Solve for the two roots using factoring or completing the square or quadratic formula.

4. The roots have three possible cases,

Case 1: Real and Distinct Roots (r1 & r2):

General Solution:

Case 2: Real and Equal Roots (r):

General Solution:

Case 3: Real and Imaginary /Complex Conjugate ( )

General Solution:

( )

Euler-Cauchy Equations

The Cauchy–Euler equation appears in a number of physics and engineering applications, such

as when solving Laplace's equation in polar coordinates.

Solution:

1. Substitute,

( )

2. The equation becomes ( ) .

3. Solve for the two roots using factoring or completing the square or quadratic formula.

4. The roots have three possible cases,

Case 1: Real and Distinct Roots (r1 & r2):

General Solution:

Case 2: Real and Equal Roots (r):

General Solution:

Case 3: Real and Imaginary /Complex Conjugate ( )

General Solution:

[ ( ) ( )]

Non-constant coefficients with missing y-term

( ) ( )

Solution:

1. Convert the equation into first order linear differential equation by substituting,

2. The equation simplifies to,

( ) ( )

3. Solve the equation using the integrating factor method.

Non-Homogeneous Second-order Linear Differential Equation

( ) ( ) ( )

where: ( )

Solution:

1. Set ( ) , the equation becomes ( ) ( ) .

2. Solve the general solution of the homogeneous differential equation ( )

3. Find the particular solution using the different method.

- Methods of Undetermined Coefficients

- Variation of Parameters

- Reduction of order

4. The general solution of the Non-homogeneous 2nd order Linear D.E. is

Methods of Undetermined Coefficients

After solving the general solution of the equivalent homogeneous differential equation

from the non-homogeneous equation, this method can now be applied in solving the particular

solution based on the function g(x).

( ) Replace y by:

Substitute the equivalent function of y, y’, and y” and solve for the arbitrary constant. The

equivalent function together with the arbitrary constant is the particular solution . This

method is limited for some forms of ( ).

Variation of Parameters

By: Engr. Efren O. Barabat