Solve the following Differential Equations

Solve the following Differential Equations 
`(x^4 +y^4)dx - 2x^3 ydy = 0`

Solution:

This is a separable differential equation, which means we can separate the variables x and y on opposite sides of the equation and then integrate them.

First, we'll move all the x terms to one side and all the y terms to the other:

(x^4 + y^4) dx = 2x^3 y dy

Next, we'll divide both sides by the coefficient of dx, which is (x^4 + y^4):

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

Now we can integrate both sides:

∫ dx/ (x^4 + y^4) = ∫ dy/y

The left side can be evaluated using the standard integral formula for (1/x^n) which is (1/(n-1))*(x^(1-n))

∫ dx/ (x^4 + y^4) = (1/3) (x^(-3) + C)

The right side is easy to integrate:

∫ dy/y = ln|y| + C

Now we have the general solution to the differential equation:

(1/3) (x^(-3) + C) = ln|y| + C

To solve for y, we can combine the constants and simplify:

(1/3) x^(-3) = ln|y| + C

y = e^((1/3)x^(-3) + C)

Note that the constant of integration C can be replaced with any constant. So, the solution is

 y = e^((1/3)x^(-3) + C)