This is a first-order homogeneous differential equation, which has the general form:
dy/dx + P(x)y = 0
In order to solve it, we will first rearrange the equation to match this form:
dy/dx - (2x-8x^2) y = 0
Now, we will apply an integrating factor. The integrating factor is typically expressed as e^(∫P(x)dx).
We calculate the integral of P(x), which is - (2x-8x^2):
∫(2x - 8X^2) dx = x^2 - 8x^3/3 + C
where C is the constant of integration. The integrating factor becomes:
e^(x^2 - 8x^3/3 + C)
We multiply every term in the differential equation by the integrating factor:
e^(x^2 - 8x^3/3 + C) * dy/dx - e^(x^2 - 8x^3/3 + C) *(2x-8x^2)y = 0
Using the property of the derivative of a product of two functions, we express the left hand side as the derivative with respect to x of [e^(x^2 - 8x^3/3 + C) * y]:
d/dx [e^(x^2 - 8x^3/3 + C) * y] = 0
Now, we can solve this differentional equation by direct integration:
∫d/dx [e^(x^2 - 8x^3/3 + C) * y] dx = ∫0 dx
e^(x^2 - 8x^3/3 + C) * y = γ
where γ is the constant of integration.
Finally, expressing y in terms of x, we get the general solution:
y(x) = γ * e^(-x^2 + 8x^3/3 - C)