Linear Differential Equation having constant coefficient:

Differential equation of the form  n is  called   as linear  differential equation   having  constant  coefficients.

In order to solve this differential equation, take auxiliary equation as a₀Dⁿ + a₁D^n-1 +...+ a_n =0

Find out the roots of this equation and then solution of given differential equation will be as given in following table

Theorem:

If  y = f₁(x) is general solution of 

and  y = f₂(x)  is a solution of

Then y = f₁(x) + f₂(x) is general solution of 

Expression f₁(x)  is called as complementary function and the expression f₂(x) is called as specific integral. These can be denotes by C.F.  and P.I. respectively.

The n^th derivative of y can be denoted Dⁿy where D stands for d/dx and n denotes the order of derivative.

If we take the Differential Equation:

then we can write this differential equation in a symbolic form as

Dⁿy + P₁D^{n- 1}y + P₂D^{n- 2}y + ....+ P_ny = X

(Dⁿ + P₁D^{n- 1} + P₂D^{n- 2} + ....+ P_n)y = X

The operator Dⁿ + P₁D^{n- 1} + P₂D^{n- 2} + ....+ P_n­ can be denoted by f(D) so that the equation takes the form f(D)y = X

y = 1/f(D) x

Methods of finding P.I. :

Notes:            

If m₁ and m₂ both are constants, the expressions (D - m₁)(D - m₂)y and (D- m₂)(D - m₁)y are equivalent that is the expression is independent of the order of operational factors.

            We will explain method with the help of following example

Example: Solve .

Solution:         The equation can be given as (D² - 5D + 6)y = e^3x

                        (D - 3) (D - 2)y = e^3x

                        Into partial functions

Email based Linear Differential Equation having constant coefficient Assignment Help - Homework Help

We at www.expertsmind.com offer email based Linear Differential Equation having constant coefficient assignment help - homework help and projects assistance from k-12 school level to university and college level and engineering and management studies. We provide finest service of Mathematics assignment help and Mathematic homework help. Our experts are helping students in their studies and they offer instantaneous tutoring assistance giving their best practiced knowledge and spreading their world class education services through e-Learning program.

Expertsmind's best education services

• Quality assignment-homework help assistance 24x7 hrs
• Best qualified tutor's network
• Time on delivery
• Quality assurance before delivery
• 100% originality and fresh work