The given differential equation is a first-order homogeneous linear differential equation of the form dy/dx = xy.

To solve this equation, we can use the technique of separation of variables. We'll rewrite the equation by dividing both sides by y and multiplying both sides by dx:

dy/y = x dx

Now, we can integrate both sides of the equation:

∫(dy/y) = ∫(x dx)

Integrating the left side gives us:

ln|y| = (1/2)x^2 + C1

where C1 is the constant of integration.

Next, we'll solve for y by taking the exponential of both sides:

|y| = e^((1/2)x^2 + C1)

Since y can be positive or negative, we'll write the solution as:

y = ± e^((1/2)x^2 + C1)

Now, let's solve for the constant C1 using the initial condition y = 1/3 when x = 2/3.

Substituting these values into the solution equation, we get:

1/3 = ± e^((1/2)(2/3)^2 + C1)

1/3 = ± e^(2/9 + C1)

Taking the natural logarithm of both sides to eliminate the exponential, we have:

ln(1/3) = (2/9 + C1)

Simplifying further, we get:

ln(1/3) = 2/9 + C1

C1 = ln(1/3) - 2/9

C1 = ln(1/3) - 2/9

Now, substituting this value of C1 back into the solution equation, we have:

y = ± e^((1/2)x^2 + ln(1/3) - 2/9)

Simplifying further, we get:

y = ± e^((1/2)x^2) / 3e^(2/9)

This is the solution to the given differential equation with the initial condition y = 1/3 when x = 2/3.