Solving the Differential Equation (4xy + 3y^4)dx + (2x^2 + 5xy^3)dy = 0
This article will guide you through the process of solving the given differential equation:
(4xy + 3y^4)dx + (2x^2 + 5xy^3)dy = 0
We'll start by analyzing the equation and then apply the appropriate method to find its solution.
Identifying the Type of Differential Equation
The given equation is a firstorder homogeneous differential equation. This is because:
 It is of the form M(x, y)dx + N(x, y)dy = 0, where M and N are functions of x and y.
 The functions M and N are homogeneous of the same degree. In this case, both M and N are homogeneous of degree 2.
Solving the Homogeneous Differential Equation
To solve this equation, we'll follow these steps:

Substitute y = vx: This substitution helps to transform the equation into a separable differential equation.

Differentiate y = vx: This will give us dy = vdx + xdv.

Substitute y and dy in the original equation: After substituting, the equation will be in terms of x, v, and dv/dx.

Separate the variables: Rearrange the terms to have all the v terms on one side and all the x terms on the other side.

Integrate both sides: Integrate both sides of the equation with respect to their respective variables.

Substitute back v = y/x: Finally, replace v with y/x to obtain the solution in terms of x and y.
Applying the Steps
Let's apply these steps to our given equation:

Substitute y = vx: (4x(vx) + 3(vx)^4)dx + (2x^2 + 5x(vx)^3)(vdx + xdv) = 0

Simplify: (4x^2v + 3x^4v^4)dx + (2x^2 + 5x^4v^3)(vdx + xdv) = 0

Expand and rearrange: (4x^2v + 3x^4v^4 + 2x^2v + 5x^4v^4)dx + (2x^3 + 5x^5v^3)dv = 0 (6x^2v + 8x^4v^4)dx + (2x^3 + 5x^5v^3)dv = 0

Separate the variables: (2x^3 + 5x^5v^3)dv = (6x^2v + 8x^4v^4)dx (2 + 5x^2v^3)dv =  (6v + 8x^2v^4)dx (2 + 5x^2v^3) / (6v + 8x^2v^4) dv = dx

Integrate both sides: The integral on the left side is a bit tricky. It might require partial fraction decomposition or substitution to solve.

Substitute back v = y/x: After integrating and obtaining the solution in terms of x and v, substitute back v = y/x to get the final solution in terms of x and y.
The Final Solution
The final solution will be a implicit equation relating x and y. It might be difficult to express it explicitly as a function of y in terms of x or vice versa.
Note: The actual integration process and the final solution will depend on the specific techniques used to solve the integral on the left side of the equation after separation.