You made a slight mistake with the variable seperation--- you can't have dx in the denominator! (Atleast, I've learned not to do it)EP said:xdv/dx=(1-4v^2)/3v
I used separation of variables to get
x/dx=(1-4v^2)/3v dv
I'm not sure if that's even right.
But if it is right, how do I integrate that?
The separation of variables method is used to solve differential equations that can be written in the form of two separate functions multiplied together. This method involves separating the variables, solving each function separately, and then combining the solutions to find the general solution.
The separation of variables method works by assuming that the solution to the differential equation can be expressed as a product of two functions, one with only the independent variable and the other with only the dependent variable. These functions are then substituted into the differential equation, and the resulting equation can be solved by integrating both sides and applying initial conditions.
The separation of variables method is typically used to solve first-order ordinary differential equations, although it can also be applied to some second-order equations. It is particularly useful for equations that can be written in the form of a separable first-order equation, such as linear, homogeneous, and exact equations.
While the separation of variables method can solve many types of differential equations, it may not work for all equations. Some equations may not be separable, and others may require more advanced techniques to solve. Additionally, the method may not always yield a closed-form solution, and numerical methods may be needed to approximate the solution.
The separation of variables method is a straightforward and intuitive approach to solving differential equations. It can also be easily applied to a wide range of equations and does not require any advanced mathematical knowledge. Additionally, the resulting solutions can often be expressed in a closed form, making them easier to interpret and analyze.