Find the particular solution of the differential equation (x^2/(y^2−2))(dx/dy)=1/(2y)?

You have:

(x^2/(y^2−2))(dx/dy)=1/(2y)

(x^2)*dx/dy = (y^2 - 2)/(2y)

This is a separable, (first-order) differential equation. Separating the variables:

x^2 dx = (y/2 - 1/y) dy

Integrate both sides:

(x^3)/3) = (y^2)/4 - ln(y) + c

where c is the constant of integration.

x^3 = 3(y^2)/4 - 3*ln(y) + C

where C = 3c is just another way of writing the constant.

This is a general, implicit solution for y(x), and an explicit solution for x(y). There is no way to obtain an explit, closed-form solution for y(x) that can be expressed in terms of elementary functions.

Now use the initial condition to find the solution to the initial value problem:

If y(1) = 3^(1/3) then:

1^3 = 3*(3^(2/3))/4 - 3*ln(3^(1/3)) + C

1 = 3/4 - ln(3) + C

C = ln(3) + 1/4

So:

x^3 = 3(y^2)/4 - 3*ln(y) + ln(3) + 1/4

x^3 = (3/4)*y^2 + ln(3/y^3) + 1/4

x(y) = [(3/4)*y^2 + ln(3/y^3) + 1/4] ^(1/3)