solve xdx/y²=dy/xz=dz/y²
Answers
Answer:
oduction. Here we shall study the methods of solving simultaneous equations of the
first order and of the first degree in the derivatives. Here we consider equations involving
only three variables. The method of solution presented here can be applied to equations
involving any number of variables.
The general type of a set of simultaneous equations of the first order having three variables
is
P1dx + Q1dy + R1dz = 0, P2dx + Q2dy + R2dz = 0,
where the coefficients are functions of x, y, z. Solving these equations simultaneously, we
have
dx
Q1R2 − Q2R1
=
dy
R1P2 − R2P1
=
dz
P1Q2 − P2Q1
,
which is of the form
dx
P
=
dy
Q
=
dz
R
,
where P, Q, and R are functions of x, y, z. Thus we note that the simultaneous equations
(1) can always be put in the form (2).
Method - I for solving dx
P
=
dy
Q
=
dz
R
....(1)
By equating two of the three fractions of (1), we may be able to get an equation in
only two variables. Some times such an equation is obtained after cancelation of some
factor from the chosen two fractions of (1). On integrating the differential equation in
only two variables by well known methods, we shall obtain one of the relations in the
general solution of (1). This method may be repeated to give another relation with help
of two other fractions of (1).
Example. Solve dx
yz
=
dy
zx
=
dz
xy
.
Solution. Taking the first two fractions, we have
xdx = ydy or 2xdx − 2ydy = 0 so that x
2 − y
2 = c1.
Again, taking the first and the third fractions, we have
xdx = zdz or 2xdx − 2zdz = 0 so that x
2 − z
2 = c2.
Therefore the general solution is given by the relation x
2 − y
2 = c1 and x
2 − z
2 = c2.
Step-by-step explanation:
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