Page 239 - DMTH504_DIFFERENTIAL_AND_INTEGRAL_EQUATION

Differential and Integral Equation

Notes Self Assessment

3. Find the characteristics of the equation
pq = z,
2
and determine the integral surface which passes through the parabola x = 0, y = z.
15.3 Summary

 Cauchy’s problem is the question to be asked, if the given differential equation solution
exists.

 The conditions are given in which the solution does exist.
 Cauchy’s characteristics equations are set up which help in the solution of the partial
differential equations.

15.4 Keywords

Depending upon the values of the parameters the solution of a particular partial differential
equation represents various integral surfaces as well as certain curves.
The characteristic method of Cauchy helps in finding a particular solution passing through
certain curves or surfaces.

15.5 Review Questions

1. Eliminate b and c from the equation
2
z = b (x + y) + b xy + c
2. Eliminate the function from the equation
2
2
2
2
(x y , x z ) = 0
Answers: Self Assessment
1. pq = z
2
2
2. yp xq + x y = 0
2t
2
t
t
3. x = 2 (e 1), y = 1/2 (e + 1), z = e , 16z = (4y +x) 2
15.6 Further Readings

Books Piaggio H.T.H., Differential Equations
Sneddon L.N., Elements of Partial Differential Equations

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