Page 156 - DMTH504_DIFFERENTIAL_AND_INTEGRAL_EQUATION

Unit 8: Total Differential Equations, Simultaneous Equations

Let the given equation (1) be identical to Notes
Adu Bdv = 0 ...(7)
From (5) du = dz.
From (6) and (7) we have

Adz Bd (xy x 2 y 2 ) = 0
or Adz ( B xdy ydx 2xdx 2ydy ) = 0 ...(8)

Rearranging in (8) we have
(By 2xB )dx (x 2 )Bdy Adz = 0 ...(9)
y
Comparing (9) with (1) we have
i
By 2xB = 2xz yz , .e B z u ...(10)
And A = xy x 2 y 2 v ...(11)

Hence (7) gives
vdu udv = 0 ...(12)
Integrating (12)
du dv
= 0
u v
or logu log v = constant = log c (say)

Therefore
u
= c
v
z
or 2 2 = c
xy x y
is the solution of equation (1).

Self Assessment

9. Solve

(a z )(ydx xdy ) xydz 0
10. Solve

(y 2 yz z 2 )dx (z 2 zx x 2 )dy (x 2 xy y 2 )dz 0

8.4 Simultaneous Differential Equations

In the unit 5 we have discussed differential equations involving two variables i.e. one independent
variable and another dependent variable. There is quite a lot of situations in which we have to
deal with a number of dependent variables that depend on one independent variable. In the
above sections also we have been dealing with more than two variables. So in these cases we can
take one variable as independent and solve the equations for the other remaining variables. We
illustrate these by means of examples.

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