Page 274 - DMTH504_DIFFERENTIAL_AND_INTEGRAL_EQUATION

Unit 17: Lagrange’s Methods for Solving Partial Differential Equations

17.6 Keywords Notes

The geometrical interpretation of the Lagrange’s equation
Pp + Qq = R

where P, Q and R are functions of Z, is that the normal to a certain surface is perpendicular to a
line whose direction cosines are in the ratio P : Q : R.
The subsidiary equations help us in finding the solution of Lagrange’s equation. If u = a, v = b
where u, v are functions of x, y, z and a, b being arbitrary constants but the statement that (u, v)
are solutions of the Lagrange equations.

17.7 Review Questions

1. Solve the following x (y z) p + y (z x) q (x y) z = 0

2. Solve the following p + q = z/a
3. Solve the following by Lagrange’s method xzp yzq = xy
2
2
4. p + q = x + y
5. zp = x
2 3
6. p q = 1
Answers: Self Assessment

1. (x + y + z) = (xyz)

1 1 1 1
2.
x y x z
3. z = e y/a f (x y)

2
2
2
4. [y + x, log (x + y + 2xy + z ) 2x] = 0
y x
5. xyz 3u = ,
x z
sin z sinx
6. f
sin y sin y

7. z = ax + (m 2 a 2 )y + c

k
8. z = ax + y + c
a
a 2
9. z = ax + n n 4 y + c
2
2
10. z = ax + 1 a y + c

11. z = ax + by + ab

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