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
first of all consider the differential equation in which the second member viz
Q is zero
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).
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
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.
us consider the equation (i),in which the second member viz. Q is also zero.If
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) giveszero and that of?(x)
for y gives Q,as y=?(x)
is solution of (i).
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.).
About my self
I am owner of Mathematics Satyam website.I am satya narain kumawat from manoharpur district-jaipur (Rajasthan) India pin code-303104.My qualification -B.SC. B.ed. I have read about m.sc. books,psychology,philosophy,spiritual, vedic,religious,yoga,health and different many knowledgeable books.I have about 15 years teaching experience upto M.sc. ,M.com.,English and science.
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