Find the general solution of the differential equation (regarding x as depending

Question

Find the general solution of the differential equation
(regarding x as depending variable and y as independent). Primes denote
derivatives with respect to x:

y'=2(xy'+y)y^(3)

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Explanation / Answer

Following is a very simple differential equation applicable to exponential decay such as the decay of radioactive element: dy(t) / dt = -k y(t) ---------- (1) where the quantity of the material y(t) is a function of the time t only, dy(t) = y(t2) - y(t1) is an infinitesimal change of y in an infinitesimal time interval dt = t2 - t1 with t2 > t1. In plain language, it states that the reduction of y, i.e., "dy" (and hence the minus sign) over a time interval "dt" depends on the remaining amount of y. The parameter k is a proportional constant which is called the rate of decay in this case, and is unique to a particular system. In particular, if k = 0, then y is not decaying at all. The concept of differentiation depends critically on the fact that a small number such as 0.00000006 divided by another small number such as 0.00000003 yields a finite value 2 (in this example). Thus dy and dt may each be infinitesimal, but the derivative dy/dt is finite. Graphically, dy/dt measures the slope of a curve as shown in Figure 02. The "d" in calculus always signifies an infinitesimal change.

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