Consider the equation dy dx = y - 4x x-y which is not separable. (a). Show that the above equation can be rewritten as dy = (y/x) - 4 dx 1 – (y/x) • - (b). Introduce a new dependent variable v so that v = y/x, or y = xv(x). Express dy/dx in terms of x, v, and dv/dx. (c). Replace y and dy/dx in the equation part (a) by the expressions from part (b) that involve v and do/dx. Show that the resulting differential equation of v is separable, and solve it. (d). Find the solution y(x) by replacing o by y/x in the solution in part (c).
Consider the equation dy dx = y - 4x x-y which is not separable. (a). Show that the above equation can be rewritten as dy = (y/x) - 4 dx 1 – (y/x) • - (b). Introduce a new dependent variable v so that v = y/x, or y = xv(x). Express dy/dx in terms of x, v, and dv/dx. (c). Replace y and dy/dx in the equation part (a) by the expressions from part (b) that involve v and do/dx. Show that the resulting differential equation of v is separable, and solve it. (d). Find the solution y(x) by replacing o by y/x in the solution in part (c).
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Transcribed Image Text:Consider the equation
dy
dx
=
y - 4x
x-y
which is not separable.
(a). Show that the above equation can be rewritten as
dy
=
(y/x) - 4
dx 1 – (y/x)
•
-
(b). Introduce a new dependent variable v so that v = y/x, or y = xv(x). Express dy/dx
in terms of x, v, and dv/dx.
(c). Replace y and dy/dx in the equation part (a) by the expressions from part (b) that
involve v and do/dx. Show that the resulting differential equation of v is separable, and
solve it.
(d). Find the solution y(x) by replacing o by y/x in the solution in part (c).
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