An A lternative M ethod for Solving Lagrange’s First-O rder Partial D ifferen tial Equation w ith Linear Function Coeffi cien ts

An A lternative M ethod for Solving Lagrange’s First-O rder Partial D ifferen tial Equation w ith Linear Function Coeffi cien ts

.A n alternativem ethod of solv ing Lagrange’s fi rst-o rder partial d ifferen tial equation of the fo rm

w here p= ?z/?x,q= ?z/?y and ai,bi,ci(i=1,2,3)are all realnum bers has been p resented here.

AM SSub ject C lassifi cations:35A 25,35A 99

Chinese Library Classifi cations:O 175.2

Lagrange’s fi rst-order partial d ifferential equation;linear functions;sim u ltaneous ord inary d ifferentialequations;linear hom ogeneousalgebraic equations.

1 In troduction

In the paper[1]by S.M.H.Islam and J.Das,am ethod of solving the partial d ifferential equation of the form

w here p= ?z/?x,q= ?z/?y and ai,bi,ci(i=1,2,3)are all real num bers,has been d iscussed.The p resent paper com p rises a detailed d iscussion of an alternativem ethod of the sam e.Thism ethod enables us to find the solu tions in som e cases of failure of the m ethod adop ted in the paper[1].

2 Them ethod

The sim u ltaneousord inary d ifferentialequations correspond ing to the PDE(1.1)are

Suppose that it is possible to find num bers ρ,αi, βi,γi(i=1,2,3) ∈ C such thateach ratio of(2.1)equals to

w here

and d D denotes the totalderivative of D:

Clearly(2.2)holds if

It is noticed from(2.3)that

The rest of the equations of(2.3)lead to the follow ing system of linear hom ogeneous algebraic equations in the unknow ns α1,β1,β2,γ1,γ2,γ3:

Let Bρdenotes the coeffi cientm atrix correspond ing to the system(2.5)i.e.

The linear hom ogeneous system(2.5)w illhave a non-trivial solu tion for(α1, β1, β2, γ1, γ2, γ3)if the determ inantof the coeffi cien tm atrix is zero,i.e.

This leads to an equation in ρ of the form:

Let

and t1=a1?ρ,t2=b2?ρ,t3=c3?ρ.Then

This im p lies that

Now

w here

Hence

Thus

Thus

So w ehave the follow ing theorem:

Theorem 2.1.Ifρ0isarootofthe equation

then 2ρ0isa rootofthe equation

Here

w here t1=a1?ρ,t2=b2?ρ,t3=c3?ρ.

Differentiating(2.12)w ith respect to ρ w eget

w hich im p lies that

Again

Hence from(2.11)w e get

w h ich im p lies that

D ifferen tiating(2.17)w ith respect to ρ w e get

w hich im p lies that

From(2.18)-(2.22)w e arriveat the follow ing theorem s:

Theorem 2.2.Ifρ0isa root oforder two ofthe equation

then 2ρ0isa rootoforder three oftheequation

Theorem 2.3.Ifρ0isarootoforder three ofthe equation

then 2ρ0is a root oforder six ofthe equation

Theorem 2.4.Ifρ1isarootoforder two and ρ2issimple rootofthe equation

then ρ1+ρ2isa root oforder two ofthe equation

Proof.Here

Hence

w here

From(2.11)w ehave

Th is im p lies that

Now,using(2.23),w eget

So,from(2.24),w e get

A lso,using(2.23)and(2.26)in(2.25),w eget

Thusw e see that(ρ1+ρ2)is a rootoforder tw o of the equation

This com p letes the p roofof the theorem.

Exam p le 2.1.Consider

For this PDE,0 is the rootoforder three of the equation

So by theorem(2.3),0 is the rootoforder six of the equation

So each ratio of(2.2),for the PDE(2.29),is equal to

So the requ ired solu tion of the PDE(2.29)is given by

w here F is an arbitrary real-valued function of tw o realvariables.

Exam p le 2.2.Consider 3

4

For this PDE,the roots of the equation

are 1,?1/2,?1/2.So by Theorem s 2.1,2.2 and 2.4 w e see that the roots o f the equation

are 2,?1,?1,?1,1/2,1/2.

So each ratio of(2.2),for the PDE(2.30),is equal to

This im p lies that

Being unable to find another solu tion of the sim ultaneous equations(2.1),in the p resent case,them ethod described above fails to derive the required solu tion of the PDE(2.30).

Exam p le 2.3.Consider

For this PDE,the equation

has on ly one root 1 of order three.Hence by theorem(2.1),2 is a root order six o f the equation

So each ratio of(2.2),for the PDE(2.31),is equal to

Being unable to find another solu tion of the sim u ltaneous equations(2.1),in the p resent case,them ethod described above fails to derive solu tion of the PDE(2.31).

Exam p le 2.4.Consider

For this PDE,theequation

has on ly one root1 of order three.Hence by Theorem(2.1),the equation

hasonly one root2 order six.For ρ =2,

So each ratio of(2.2),for the PDE(2.32),is equal to

Again w e see that the sim u ltaneous ord inary d ifferential equations(2.1),for the PDE (2.32),are equal to

Thusw ehave

From this relationsw ehave the required solution of the PDE(2.32)as

w here F is an arbitrary real-valued function of tw o realvariables.

Exam p le 2.5.Consider

Here it can be show n that the equation

has roots 0 oforder 1,?6 oforder 2 and ?12 oforder 3.

So the ratios in(2.2),for the PDE(2.33),becom e

This im p lies that

So w e get on ly one solu tion o f the sim u ltaneous equations(2.1),in the p resent case. Hence them ethod described above fails to derive the required solu tion of the PDE(2.33).

ISLAM Syed M d H im ayetu l1,,DASJ.21Fatu llapur Adarsha H igh School,Vill.-Fatullapur,P.O.-N im ta,D ist.-North 24-Parganas,Kolkata-700049,India.
2Department ofPureMathematics,CalcuttaUniversity,35 Ballygunge
Circu larRoad,Kolkata-700019,West Bengal,India.
Received 19M arch 2015;A ccep ted 30 Ju ly 2015

.Email addresses:him u2000@yahoo.com(S.M d H.Islam),j tdas2000@yahoo.com(J. Das)

[1]Islam S.M.H.,Das J.,Am ethod ofsolving Lagranges fi rst-order partiald ifferentialequation w hose coeffi cientsare linear functions.IJDEA,14(2)(2015),65-79.