Problems in Mathematical Analysis.pdf - pwp.net.ipl.pt

212 Functions of Several Variables [Ch. 6
\. Change of variables in expressions containing ordinary derivatives.
Example 1. Transform the equation
2 y
dx x
putting * = y-
Solution. Express the derivatives of y with respect to x in terms of the
derivatives of y with respect to /. We have
dt
dy dy
dt dt
dx dx _
dt t*
Substituting the expressions of the derivatives just found into the given
equation and replacing x by -r- , we get
or
dt* ruij -"'
Example 2. Transform the equation
x g + ^y_g =0 .
taking y for the argument and x for the function.
Solution. Express the derivatives of y with respect to x in terms of the
derivatives of x with respect to y.
\
^ -
dx^Jx''
dy
dx* dx{ dx } dyl dx \dx~~ fdx\ 2
dt '
'
-
dx "~
fdx\* '
\ dy (Ty)
J dy (dy)
Substituting these expressions of the derivatives into the given equation, we
will have
d*x
dj
dy
1 L=o '
' dx
\dy dy