Page 162 - DMTH504_DIFFERENTIAL_AND_INTEGRAL_EQUATION

Unit 8: Total Differential Equations, Simultaneous Equations

Notes
1 c
c
or y = log(z 2 ) 2
1
c c
1 1
c
or c y = log(z 2 ) c 2 ...(6)
1
1
Substituting value of c from (3)
1
2x
x = log z c 2 ...(7)
y
Thus from (3), (7) we have
x
c =
1
y
...(8)
zy 2x
c = x log
2 y
So equation (8) form the complete integral of the set of equations.

Self Assessment

13. Solve

dx dy dz
1 y 1 x z

14. Solve
dx dy dz
x 2 y 2 yz x 2 y 2 yz ( z x y )
Geometrical Meaning of

dx dy dz
= ...(1)
P Q R

We know that the direction ratio of the tangent to a curve at any point (x, y, z) on it are proportional
,
,
to dx dy dz at that point. Hence geometrically the given equations represent a system of
curves in space, such that the direction ratios of the tangent to any one of these curves in space,
at that point ( , , )x y z on it are proportional to P, Q and R at that point. If u , a v b are the
general solutions of (1), then system of curves must be the curves of intersection of the surfaces
u , a v . b It is also clear that since a, b are arbitrary constants, the system of curves represented
by the equations is doubly infinite.

8.5 Summary

 Total differential equations can be solved under certain conditions.

 Simultaneous Differential equations are also shown to be solved by the above method.
 Illustrated examples are solved so that the technique of solving by various methods is
clear.

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