# Linear Differential Equations

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```					                 Linear Differential Equations
Linear Differential Equations

The general representation of Higher Order Linear Differential equations of constant
coefficients is

yn + an-1 ( x ). yn-1 + an-2 ( x ). yn-2 +… + a0 ( x ). y = g ( x )   (1)

The general form of nth order linear differential equation

an(x)yn + an-1(x). yn-1 + an-2(x). yn-2 +… + a0(x). Y = g ( x )      (2)

And it can be rewritten as

ym = dmy / dxm                                                         (3)

Where an ( x ), an-1 ( x )…a0 ( x ) are the continuous functions of x. if g ( x ) = 0 then the
equation is called as homogenous differential equation. If g ( x ) ≠ 0 then this equation is
called as non homogenous differential equation.

Some theorems are there to understand higher order differential equations better.

Know More About Solving Equations with Variables on both Sides

Math.Tutorvista.com                                                        Page No. :- 1/4
Theorem 1:

Assume the functions a0, a1, …, an-1 and g(t) are all continuous in some open interval I
containing x0 then there is a unique solution provided to the differential equation given by the
equations above defined and the solution will exist for all t and I.

Let’s have a homogenous equation of Higher Order Linear Differential equations as below

yn + an-1 ( x ). yn-1 + an-2 ( x ). yn-2 +… + a0 ( x ). y = 0                (4)

Assume that y1 ( x ), y2 ( x ), … , yn ( x ) are the solution of the above homogenous equation
the by the use of principle of superposition method

y( x ) = c1 y1( x ) + c2 y2 ( x ) + … + cnyn( x )                      (5)

The expression above written is also will be a solution of the homogeneous differential
equation.

Then the value of the constants c1, c2, … , cn for any value of x0( as defined in theorem 1)
can be easily calculated

c1 y1 (x0) + c2 y2 ( x0 ) + … + cnyn ( x0 ) = ?o ,

c1 y1‘(x0) + c2 y2‘( x0 ) + … + cnyn’( x0 ) = ?1 ,

:

:

c1 y1(n-1) (x0) + c2 y2(n-1) ( x0 ) + … + cnyn(n-1) ( x0 ) = ?n-1,

Math.Tutorvista.com                                                        Page No. :- 2/4
Theorem 2:

Assume functions a0, a1, …, an-1 and g(t) are all continuous in some open interval I and also
assume that y1( x ) , y2 ( x ), … , yn ( x ) form a fundamental set of solutions and the general
solution of equation 4 as defines above is

y(x) = c1 y1 (x0) + c2 y2 ( x0 ) + … + cnyn ( x0 ) ,

Theorem 3:

Assume that Y1(x) ,Y2(x) are two solutions for equation 1 and that y1 ( x ), y2 ( x ),…., yn ( x )
are a fundamental set of solutions to the homogenous differential equation 4 the Y1 ( x ) – Y2
( x ) would be a solution for the equation 4 and can be written in the form

Y1(x) – Y2(x) = c1 y1 ( x) + c2 y2 ( x ) + … + cnyn ( x )

Math.Tutorvista.com                                                      Page No. :- 4/4
Thank You

Math.TutorVista.com

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