a. Show that the general solution to the differential equation d y d x = x + a y a x − y can be written in polar form as r = k e a θ . b. For the particular case when a = 1 2 , determine the solution satisfying the initial condition y ( 1 ) = 1 , and find the maximum x -interval on which this solution is valid. c. On the same set of axes, sketch the spiral corresponding to your solution in (b), and the line y = x 2 . Thus verify the x -interval obtained in (b) with the graph. d.
a. Show that the general solution to the differential equation d y d x = x + a y a x − y can be written in polar form as r = k e a θ . b. For the particular case when a = 1 2 , determine the solution satisfying the initial condition y ( 1 ) = 1 , and find the maximum x -interval on which this solution is valid. c. On the same set of axes, sketch the spiral corresponding to your solution in (b), and the line y = x 2 . Thus verify the x -interval obtained in (b) with the graph. d.
Solution Summary: The author explains the general solution of the differential equation dy dx, which can be written in polar form.
a. Show that the general solution to the differential equation
d
y
d
x
=
x
+
a
y
a
x
−
y
can be written in polar form as
r
=
k
e
a
θ
.
b. For the particular case when
a
=
1
2
, determine the solution satisfying the initial condition
y
(
1
)
=
1
, and find the maximum
x
-interval on which this solution is valid.
c. On the same set of axes, sketch the spiral corresponding to your solution in (b), and the line
y
=
x
2
. Thus verify the
x
-interval obtained in (b) with the graph.
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