# Linear Equations With Constant Coefficients

__LinearEquations With Constant Coefficients__
##
Linear Equations With Constant Coefficients |

*Definition-**Alinear differential equation** is an equation in which the dependent variable y
and its differential coefficients occur only in the first degree. The general
form of such an equation is*

*(dny/dxn*

)+P1(dn-1y/dxn-1)+P2(dn-2y/dxn-2)+……..+Pny=0 ……(i))+P1(dn-1y/dxn-1)+P2(dn-2y/dxn-2)+……..+Pny=0 ……(i)

*Where Q*

and P1,P2,P3,…..,Pn are all

constants or functions of x.and P1,P2,P3,…..,Pn are all

constants or functions of x.

*If P1,P2,P3,…..,Pn*

are all constants (Q may not be

constant), then the equation is said to be a linear differential equation

constant coefficients.are all constants (Q may not be

constant), then the equation is said to be a linear differential equation

constant coefficients.

*We shall*

first of all consider the differential equation in which the second member viz

Q is zerofirst of all consider the differential equation in which the second member viz

Q is zero

*i.e. (dny/dxn*

)+P1(dn-1y/dxn-1)+P2(dn-2y/dxn-2)+……..+Pny=0 ……(ii))+P1(dn-1y/dxn-1)+P2(dn-2y/dxn-2)+……..+Pny=0 ……(ii)

*If y=f1(x)*

be a solution of (ii),then by substitution in (ii) it can be seen that y=cf1(x),where

C is an arbitrary constant,is also a solution of (ii).be a solution of (ii),then by substitution in (ii) it can be seen that y=cf1(x),where

C is an arbitrary constant,is also a solution of (ii).

*Similarly*

if y=f2(x),y=f3(x)……..,y=fn((x) are the

solutions of (ii),then y=C2f2(x),y=C3f3(x),….,y=Cnfn(x),

where C2,C3,…..,Cn are arbitrary constants,are

also the solutions of (ii) Also substitution will show thatif y=f2(x),y=f3(x)……..,y=fn((x) are the

solutions of (ii),then y=C2f2(x),y=C3f3(x),….,y=Cnfn(x),

where C2,C3,…..,Cn are arbitrary constants,are

also the solutions of (ii) Also substitution will show that

*. y=C1f1(x)+C2f2(x)+…+Cnfn(x) ………(iii)*

*Is also a*

solution of (ii).If f1(x),f2((x),f3(x),….are

linearly independent,then (iii) is the complete integral of (ii),since it

contains n arbitrary constants and (ii) is order n.solution of (ii).If f1(x),f2((x),f3(x),….are

linearly independent,then (iii) is the complete integral of (ii),since it

contains n arbitrary constants and (ii) is order n.

*Now let*

us consider the equation (i),in which the second member viz. Q is also zero.If

y=?(x)

be solution of (i),thenus consider the equation (i),in which the second member viz. Q is also zero.If

y=?(x)

be solution of (i),then

*.y=F(x)+?(x) be a solution of (i),then*

where F(x)= C1f1(x)+C2f2(x)+…+Cnfn(x) is also a solution of (i) since the

substitution of F(x) for y in the left hand member of (i) gives zero and that of ?(x)

for y gives Q,as y=?(x)

is solution of (i).where F(x)= C1f1(x)+C2f2(x)+…+Cnfn(x) is also a solution of (i) since the

substitution of F(x) for y in the left hand member of (i) gives zero and that of ?(x)

for y gives Q,as y=?(x)

is solution of (i).

*The*

solution (iv) contains n arbitrary constants and (i) is differential equation

of nth order,therefore it is the complete solution of (i). The part F(x) is

called the complementary function (C.F.) and the part ?(x) is called thesolution (iv) contains n arbitrary constants and (i) is differential equation

of nth order,therefore it is the complete solution of (i). The part F(x) is

called the complementary function (C.F.) and the part ?(x) is called the

__particularintegral (P.I.).__
##
Linear Equations With Constant Coefficients |