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(d^n-1y/dx^n-1)+P₂(d^n-2y/dx^n-2)+……..+P_ny=0          ……(i)

Where Q and P₁,P₂,P₃,…..,P_n are all constants or functions of x.

If P₁,P₂,P₃,…..,P_n  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(d^n-1y/dx^n-1)+P₂(d^n-2y/dx^n-2)+……..+P_ny=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=f_n((x) are the solutions of (ii),then y=C₂f₂(x),y=C₃f₃(x),….,y=C_nf_n(x), where C₂,C₃,…..,C_n are arbitrary constants,are also the solutions of (ii) Also substitution will show that

. y=C₁f₁(x)+C₂f₂(x)+…+C_nf_n(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=𝞍(x) be solution of (i),then

.y=F(x)+𝞍(x) be a solution of (i),then where F(x)= C₁f₁(x)+C₂f₂(x)+…+C_nf_n(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  𝞍(x) for y gives Q,as y=𝞍(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 𝞍(x) is called the particularintegral (P.I.).

Linear Equations With Constant Coefficients