Solve the differential equation below

(Y + y^3/3 + x^2/2)dx + 1/4(1 + y^2)xdy=0

To solve the given differential equation:

Step 1: Identify the type of the differential equation.
The given equation is a first-order linear ordinary differential equation.

Step 2: Rearrange the equation into standard form.
To do this, you need to group the terms involving dx and dy separately. Let's start by rearranging the equation:

(Y + y^3/3 + x^2/2)dx + 1/4(1 + y^2)xdy = 0

Multiply both sides of the equation by 4 to clear the fraction:

4(Y + y^3/3 + x^2/2)dx + (1 + y^2)xdy = 0

Expanding gives:

4Ydx + 4(y^3/3)dx + 2x^2dx + x(1 + y^2)dy = 0

Now we have the equation in standard form: M(x, y)dx + N(x, y)dy = 0, where:
M(x, y) = 4Y + 4(y^3/3) + 2x^2
N(x, y) = x(1 + y^2)

Step 3: Check for exactness.
To check for exactness, we need to verify whether the partial derivatives of M with respect to y (My) and N with respect to x (Nx) are equal.
In this case, My = 4y^2, and Nx = 2x. Since My is not equal to Nx, the equation is not exact.

Step 4: Find an integrating factor.
To make the equation exact, we need to find an integrating factor, denoted by μ(x). The integrating factor can be found using the formula:

μ(x) = e^(∫((Nx - My)/M)dx)

In this equation, M = 4Y + 4(y^3/3) + 2x^2, My = 4y^2, and Nx = 2x.

Plugging in the values, we have:

μ(x) = e^(∫((2x - 4y^2)/(4Y + 4(y^3/3) + 2x^2))dx)

Step 5: Evaluate the integral to find the integrating factor.
To find the integrating factor, you need to evaluate the integral:

μ(x) = e^(∫((2x - 4y^2)/(4Y + 4(y^3/3) + 2x^2))dx)

The integral might not have a closed-form solution, so you can use numerical methods or approximation techniques to find the value of the integral.

Step 6: Multiply the equation by the integrating factor.
Once you have the integrating factor μ(x), multiply the original equation by it:

μ(x)[4(Y + y^3/3 + x^2/2)dx + (1 + y^2)xdy] = 0

This will make the equation exact.

Step 7: Check for exactness.
After multiplying by the integrating factor, check whether the equation has become exact. If the equation is now exact, proceed to the next step. If not, recheck your calculations.

Step 8: Solve the exact equation.
If the equation has become exact, you can now solve it. The exact differential equation can be written as:

d(μ(x)M(x, y))/dx = d(μ(x)N(x, y))/dy

Integrate both sides with respect to the corresponding variable(s), and solve the resulting equation to find the solution.

Unfortunately, without the specific values of the constants Y, it is not possible to provide the exact solution to the differential equation. However, by following the steps outlined above, you should be able to solve the equation once you have the specific values.