PARTIAL DIFFERENTIAL EQUATIONS

Formation of Partial Differential equations Partial Differential Equation can be formed either by elimination of arbitrary constants or by the elimination of arbitrary functions from a relation involving three or more variables . SOLVED PROBLEMS 1.Eliminate two arbitrary constants a and b from here R is known constant .

solution (OR) Find the differential equation of all spheres of fixed radius having their centers in x y- plane. solution Differentiating both sides with respect to x and y

By substituting all these values in (1) or

2. Find the partial Differential Equation by eliminating arbitrary functions from SOLUTION

By

3.Find Partial Differential Equation by eliminating two arbitrary functions from SOLUTION Differentiating both sides with respect to x and y

Again d . w .r. to x and yin equation (2)and(3)

Different Integrals of Partial Differential Equation 1. Complete Integral (solution) Let be the Partial Differential Equation. The complete integral of equation (1) is given by

A solution obtained by giving particular values to 2. Particular solution A solution obtained by giving particular values to the arbitrary constants in a complete integral is called particular solution . 3.Singular solution The eliminant of a , b between when it exists , is called singular solution

In equation (2) assume an arbitrary relation 4. General solution In equation (2) assume an arbitrary relation of the form . Then (2) becomes Differentiating (2) with respect to a, The eliminant of (3) and (4) if exists, is called general solution

The Partial Differential equation of the form Standard types of first order equations TYPE-I The Partial Differential equation of the form has solution with TYPE-II The Partial Differential Equation of the form is called Clairaut’s form of pde , it’s solution is given by

TYPE-III If the pde is given by then assume that

The given pde can be written as .And also this can be integrated to get solution

The pde of the form can be solved by assuming TYPE-IV The pde of the form can be solved by assuming Integrate the above equation to get solution

Complete solution is given by SOLVED PROBLEMS 1.Solve the pde and find the complete and singular solutions Solution Complete solution is given by with

d.w.r.to. a and c then Which is not possible Hence there is no singular solution 2.Solve the pde and find the complete, general and singular solutions

The complete solution is given by with

no singular solution To get general solution assume that From eq (1)

Solution Eliminate from (2) and (3) to get general solution 3.Solve the pde and find the complete and singular solutions Solution The pde is in Clairaut’s form

complete solution of (1) is d.w.r.to “a” and “b”

From (3)

is required singular solution

4.Solve the pde 5.Solve the pde Solution Solution pde Complete solution of above pde is 5.Solve the pde Solution Assume that

From given pde

Integrating on both sides

6. Solve the pde Solution Assume Substituting in given equation

Integrating on both sides 7.Solve pde (or) Solution

Integrating on both sides Assume that Integrating on both sides

8. Solve the equation Solution integrating

Equations reducible to the standard forms (i)If and occur in the pde as in Or in Case (a) Put and if ;

where Then reduces to Similarly reduces to

case(b) If or put (ii)If and occur in pde as in Or in

Case(a) Put if where Given pde reduces to and

Case(b) if Solved Problems 1.Solve Solution

where

(1)becomes

2. Solve the pde SOLUTION

Eq(1) becomes

Lagrange’s Linear Equation Def: The linear partial differenfial equation of first order is called as Lagrange’s linear Equation. This eq is of the form Where and are functions x,y and z The general solution of the partial differential equation is Where is arbitrary function of and

auxilary equations are Here and are independent solutions of the auxilary equations Solved problems 1.Find the general solution of Solution auxilary equations are

Integrating on both sides

2.solve solution The general solution is given by Auxiliary equations are given by

Integrating on both sides

Integrating on both sides

HOMOGENEOUS LINEAR PDE WITH CONSTANT COEFFICIENTS The general solution is given by HOMOGENEOUS LINEAR PDE WITH CONSTANT COEFFICIENTS Equations in which partial derivatives occurring are all of same order (with degree one ) and the coefficients are constants ,such equations are called homogeneous linear PDE with constant coefficient

Assume that then order linear homogeneous equation is given by or

or Rules to find complementary function The complete solution of equation (1) consists of two parts ,the complementary function and particular integral. The complementary function is complete solution of equation of Rules to find complementary function Consider the equation or

Case 1 The auxiliary equation for (A.E) is given by And by giving The A.E becomes Case 1 If the equation(3) has two distinct roots The complete solution of (2) is given by

Rules to find the particular Integral Case 2 If the equation(3) has two equal roots i.e The complete solution of (2) is given by Rules to find the particular Integral Consider the equation

Particular Integral (P.I) Case 1 If then P.I= If and is factor of then

P.I If and is factor of then P.I Case 2 P.I

Case 4 when is any function of x and y. P.I= Expand in ascending powers of or and operating on term by term. Case 4 when is any function of x and y. P.I=

1.Find the solution of pde Here is factor of Where ‘c’ is replaced by after integration Solved problems 1.Find the solution of pde Solution The Auxiliary equation is given by

2. Solve the pde Solution The Auxiliary equation is given by By taking Complete solution 2. Solve the pde Solution The Auxiliary equation is given by

3. Solve the pde Solution the A.E is given by

4. Find the solution of pde Complete solution = Complementary Function + Particular Integral The A.E is given by

Complete solution

5.Solve Solution

6.Solve Solution

7.Solve Solution

7.Solve Solution A.E is

Non Homogeneous Linear PDES If in the equation the polynomial expression is not homogeneous, then (1) is a non- homogeneous linear partial differential equation Ex Complete Solution = Complementary Function + Particular Integral To find C.F., factorize into factors of the form

If the non homogeneous equation is of the form 1.Solve Solution

Assignment

1.Find the differential equation of all spheres of fixed radius having centre in xy-plane. 2.Solve the pde z=ax3+by3 by eliminating the arbitrary constants. 3.Solve the pde z=f(x2-y2) by eliminating the arbitrary constants. 4.solve

5 .(D²+2DD+D²)z=exp(2x+3y) 6.4r-4s+t=6log(x+2y) 7.find the general solution of differential equation (D²+D+4)z=exp(4x-y)

TEST NOTE:- DO ANY TWO

1. (D-D-1)(D-D-2)Z=EXP(2X-Y) 2 1.(D-D-1)(D-D-2)Z=EXP(2X-Y) 2.SOLVE (D3+D2D-DD2-D3)Z=EXP(X)COS2Y 3.SOLVE (X-Y)p+(X+Y)q=2XZ