This is a separable differential equation, which means that we can separate the variables x and y and then integrate both sides with respect to their respective variables.
Starting with:
dy/dx = (4x^2 +3y^2)/2xy
We can rearrange the equation to separate the variables:
(2y^2)dy = ((4x^2)/(2x))dx + ((3y^2)/(2y))dx
Simplifying this equation, we get:
y^2 dy = 2x^2 dx + (3/2) y dx
Now we can integrate both sides:
∫y^2 dy = ∫(2x^2 dx + (3/2) y dx)
Integrating the left-hand side, we get:
(1/3) y^3 + C1
where C1 is the constant of integration.
Integrating the right-hand side, we get:
(2/3) x^3 + (3/4) y^2 + C2
where C2 is the constant of integration.
Putting these two results together, we get:
(1/3) y^3 = (2/3) x^3 + (3/4) y^2 + C
where C is the constant of integration that combines C1 and C2.
Simplifying this equation, we get:
y^3 = 2x^3 + (9/4) y^2 + C
where we have absorbed the constant (1/3)C into C.
This is the general solution to the differential equation.