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.
