Home | | Transforms and Partial Differential Equations | Non Homogeneous Linear Equations

# Non Homogeneous Linear Equations

The methods for finding the Particular Integrals are the same as those for homogeneous linear equations.

Non â€“Homogeneous Linear Equations

Let us consider the partial differential equation

f (D,D') z = F (x,y)------- (1)

If f (D,D') is not homogeneous, then (1) is a nonâ€“homogeneous linear partial differential equation. Here also, the complete solution = C.F + P.I.

The methods for finding the Particular Integrals are the same as those for homogeneous linear equations.

But for finding the C.F, we have to factorize f (D,D') into factors of the form D â€“mD' â€“c.

Consider now the equation

(D â€“mD' â€“c) z = 0 -----------  (2).

This equation can be expressed as

p â€“mq = cz ---------(3),

which is in Lagrangian form.

The subsidiary equations are The solutions of (4) are y + mx = a and z = becx.

Taking b =  f (a), we get z = ecx f (y+mx) as the solution of (2).

Note:

1.  If   (D-m1D' â€“C1) (D â€“m2D'-C2)   â€¦â€¦ â€“m(Dn'-Cn) z = 0 is the partial

differential equation, then its complete solution is

z = ec1x f1(y +m1x) + ec2x f2(y+m2x) + . . . . . + ecnx fn(y+mnx)

2.  In the case of repeated factors, the equation (D-mD' â€“C)nz = 0 has a complete

solution z = ecx f1(y +mx) + x ecx f2(y+mx) + . . . . . +x n-1 ecx fn(y+mx).

Example 31

Solve (D-D'-1) (D-D' â€“2)z = e 2x â€“y

Here  m1 = 1, m2 = 1, c1 = 1, c2 = 2.

Therefore, the C.F is ex f1 (y+x) + e2x f2 (y+x). Example 32

Solve (D2 â€“DD' + D' â€“1) z = cos (x + 2y)

The given equation can be rewritten as

(D-D'+1) (D-1) z = cos (x + 2y)

Here m1 = 1, m2 = 0, c1 = -1, c2 = 1.

Therefore, the C.F = eâ€“x f1(y+x) + ex f2 (y) Example 33

Solve [(D + D'â€“1) (D + 2D' â€“3)] z = ex+2y + 4 + 3x +6y

Here m1 = â€“1, m2 = â€“2 , c1 = 1, c2 = 3.

Hence the C.F is z = ex f1(y â€“x) + e3x f2(y â€“2x).  It is the complete solution.

Exercises

(a) Solve the following homogeneous Equations. 6. (D2 + 4DD' â€“5D'2) z = 3e2x-y + sin (x â€“2y)

7. (D2 â€“DD' â€“30D'2) z  = xy + e6x+y

8. (D2 â€“4D' 2) z = cos2x. cos3y

9. (D2 â€“DD'  â€“2D'2) z = (y â€“1)ex

10.4r + 12s + 9t =  e3x â€“2y

(b)Solve the following non â€“homogeneous equations.

1.     (2DD' + D' 2 â€“3D') z = 3 cos(3x â€“2y)

2.     (D2 + DD' + D' â€“1) z = e-x

3.     r â€“s + p = x2 + y2

4.     (D2 â€“2DD' + D'2 â€“3D + 3D' + 2)z = (e3x + 2e-2y)2

(D2 â€“D'2 â€“3D + 3D') z = xy + 7.

Study Material, Lecturing Notes, Assignment, Reference, Wiki description explanation, brief detail
Mathematics (maths) : Partial Differential Equations : Non Homogeneous Linear Equations |