Question

use the differential equation given by dy/dx=xy/3, y > 0. Find the particular solution y = f(x) to the given differential equation with the initial condition f(0) = 4.

Answer 1

dy/dx=xy/3, y > 0 can be rewritten in terms of only one variable (x or y) each:

3 dy
----- = x*dx (first term is all in y; second term all in x).
y
x^2
Integrating, 3 ln y = ------- + ln C, Then ln y^3 = (1/2)x^2 + ln C.
2

Combining the log terms:

ln y^3-ln C = (1/2)x^2.

Given: if x=0, y=4. Subst. these v alues into the equation in y given above:
4^3
ln 4^3 - ln C = (1/2)(0)^2 = 0. Then ln ------- = 0, which tells us that
C

4^3 = C. Thus, the solution is ln y^3 = (1/2)x^2 + ln (4/3).

There are other ways in which you could write this same expression. You could, for example, solve for either y^3 or y alone.