Page 302 - DMTH504_DIFFERENTIAL_AND_INTEGRAL_EQUATION

Unit 19: Jacobi’s Method for Solving Partial Differential Equations

19.3 Summary Notes

 Jacobi’s method of solution of the partial differential equation of the first order is very
similar to that of Charpit’s method.

 The method consists in setting up subsidiary equations through which two integrals are
found that help in finding the solution.

19.4 Keywords

The subsidiary equations help us in finding the two independent integrals.

u u u
Independent integrals help in finding the partial derivatives , , and so the solution
x 1 x 2 x 3
can be found.

19.5 Review Questions

1. Find the solution of

F = p p p 2 3x 3x 4x 2 0
1 2 2 1 2 3
with additional equations
F = x p x p 2x 2 2x 2 0
1 1 1 2 2 1 2
F = p  2x = 0
2 3 3
2. Find complete integral of
2
p x + p 3 = 0
1 3
p x 2 p x 2 = 0
3 2
2 3
3. Find the complete integral of
2x x p p z + x p = 0
1 3 1 3 2 2
Answers: Self Assessment

1
1. z a x a x sin (a a x ) a 3
2 2
1 1
1 2 3
2. 4a z 4a 2 1 log x 3 2a a (x 1 x 2 ) (x 1 x 2 ) 2 4a a
1 2
1
1 3
3. z ( a x 1 log x 2 1/x 3 ) b
4. z x x ( a x 3 x 4 ) b
1 2
19.6 Further Readings

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

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