# Linear Equations With Constant Coefficients

LinearEquations With Constant Coefficients
Contents

#### 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)
Where Q
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.
We shall
first 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)
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).
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 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.
Now let
us 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).
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 the particularintegral (P.I.).