# Solving a non-linear non-seperable differential equation

In summary, a non-linear non-separable differential equation is a type of equation that involves derivatives of an unknown function and cannot be separated into simple, linear equations. These equations are difficult to solve because they require advanced mathematical techniques and do not have a general solution. Some common techniques for solving them include separation of variables, substitution, and using power series or numerical methods. However, not all equations can be solved analytically and may require approximation techniques. Non-linear non-separable differential equations are widely used in science to model complex systems and phenomena, make predictions, and develop mathematical models and simulations.

I'm having trouble solving this differential equation:

(2x^2)yy' = −1

I'm just really not sure how to go about solving this.

Why isn't it separable?

$$2x^2 y \frac{dy}{dx} = -1 \Rightarrow 2y dy = -\frac{dx}{x^2}$$

Ah yes, that's actually quite easy, I'm just being dense.

Thanks for the help.

## 1. What is a non-linear non-separable differential equation?

A non-linear non-separable differential equation is a type of equation that involves derivatives of an unknown function, where the derivatives are not directly proportional to the function itself and cannot be separated into simple, linear equations.

## 2. Why are non-linear non-separable differential equations difficult to solve?

Non-linear non-separable differential equations are difficult to solve because they do not have a general solution that can be applied to all cases. Each equation must be solved using different methods, and often these methods require advanced mathematical techniques and algorithms.

## 3. What are some common techniques for solving non-linear non-separable differential equations?

Some common techniques for solving non-linear non-separable differential equations include separation of variables, substitution, and using power series or numerical methods. These techniques can be combined and adapted to solve specific equations.

## 4. Can all non-linear non-separable differential equations be solved analytically?

No, not all non-linear non-separable differential equations can be solved analytically. Some equations may have no solution or only approximate solutions. In these cases, numerical methods or approximation techniques may be used to find a solution.

## 5. How are non-linear non-separable differential equations used in science?

Non-linear non-separable differential equations are used in many areas of science, such as physics, engineering, and biology. They are used to model complex systems and phenomena, and to make predictions and understand the behavior of these systems. They are also used in the development of mathematical models and simulations.