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Monges Method

This document discusses partial differential equations of second order. It begins with objectives of understanding how second order PDEs can be solved using ordinary differential equation methods. It then covers solving PDEs by inspection, Monge's method, and a variation of Monge's method. Specific examples are provided for each solution technique. References for further reading on differential equations, calculus of variations, and special functions are listed at the end.

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872 views22 pages

Monges Method

This document discusses partial differential equations of second order. It begins with objectives of understanding how second order PDEs can be solved using ordinary differential equation methods. It then covers solving PDEs by inspection, Monge's method, and a variation of Monge's method. Specific examples are provided for each solution technique. References for further reading on differential equations, calculus of variations, and special functions are listed at the end.

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Upperwala Rai
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© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
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Dr.

Kamlesh
Bisht
BOOK: Differential Equation, Calculus of (Mathematics)
Variations and Special Functions
UNIT – III : Partial Differential Equation Of
Second Order, Monge’s Method

DR. KAMLESH BISHT


Academic Consultant
Department of Mathematics
Uttarakhand Open University, Haldwani
Dr. Kamlesh Bisht
(Mathematics)

Content
 Objective
 Introduction
 Solution of P.D.E. of second order by inspection
 Solution of P.D.E. by Monge’s method
 Solution of P.D.E. by another form of Monge’s
method
 References.
Dr. Kamlesh Bisht
(Mathematics)

Objectives
After studying this unit, you should be able to-
 Discuss partial differential equations of order two
with variable coefficients.
 Learn how a large class of second order partial
differential equation may be solved by using the
methods applicable for solving ordinary differential
equation.
 Study Monge’s method for solution of some special
type of second order partial differential equation.
Introduction
A partial differential equation (P.D.E) is said to be of order two, if it involves at least
one of the differential coefficients r, s, t and none of order higher than two. The
general form of second order partial differential equation in two independent
variable x, y is given as:

The most of the general linear partial differential equation of second order in two
independent variable x and y with variable coefficient is given as:
Solution of P.D.E of second order by inspection
Before taking up the genetral equation of second degree P.D.E., we
discus the solution of simple problems which can be integrated merely
by inspection. On two successive integral of given P.D.E., we get the
general solution which is relation in x, y, z. To understand this we
discuss the following problems.

Question: Solve t+ s+ q = 0
Solution: We can write the given problem as
Question: Show that a surface passing through the circle z=0, x2+y2=1
and satisfying the differential equation s=8xy is z= (x2+y2)2-1
Solution: We can write the given differential equation as

………. (5)
….. (6)

……(7)
Monge’s Method For Solving Equation Of The Type
Rr+Ss+Tt = V
Question: Solve r = a2t by Monge’s method
Answer: Comparing the given equation with Rr+Ss+Tt=V, we get R=1,
S=0, T=-a2, V=0. The Monge’s subsidiary equation are given by
Monges’ Method For Solving Equation Of The Type
Rr+Ss+Tt+U(rt-s2)=V
Question: Solve 3r+4s+t+(rt-s2)=1
Solution: Comparing the given equation with Rr+Ss+Tt+U(rt-s2)=V, we
have R=3, S=4, T=1, U=1, V=1. Then λ- quadratic equation
References:
 Advanced Differential Equation
M.D. Raisinghania, S. Chand Publication

 SLM of VMOU, Kota


Differential Equation, Calculus of Variation and
Special Functions.
Dr. Kamlesh Bisht
(Mathematics)
Mob. No.-
8279829875

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THANKS

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