# Non linear second order diff eq

• rbetzel
In summary, a non linear second order differential equation is a mathematical equation that involves a second derivative of a function and has at least one term that is not a linear function of the dependent variable or its derivatives. It differs from a linear equation in that it cannot be solved using traditional methods and requires more advanced techniques. These equations are commonly used in fields such as science and engineering to model complex systems, but can be difficult to solve and may require numerical or approximation methods.
rbetzel
I'm trying to find a substitution that works for the nonlin diffeq

y''+a(y')^2-by=0

Any suggestions?

If you differentiate again, you get

$$y^{'''} + 2ay^{'} \cdot y^{''} - b = 0$$ right?

If you let

$$u = y^{'}$$ you get $$u^{''} + 2a u^{'}u - b = 0$$

## 1. What is a non linear second order differential equation?

A non linear second order differential equation is a mathematical equation that involves a second derivative of a function and has at least one term that is not a linear function of the dependent variable or its derivatives. This means that the equation cannot be solved using traditional methods and requires more advanced techniques.

## 2. How is a non linear second order differential equation different from a linear one?

A linear second order differential equation only involves terms that are linear functions of the dependent variable or its derivatives, making it possible to solve using traditional methods. Non linear equations, on the other hand, have at least one term that is not a linear function and require more advanced techniques such as numerical methods or approximation methods for solutions.

## 3. What are some real-life applications of non linear second order differential equations?

Non linear second order differential equations are used in many fields of science and engineering to model complex systems. Some common applications include predicting population growth, analyzing chemical reactions, and studying the mechanics of objects in motion.

## 4. How are non linear second order differential equations solved?

There are several methods for solving non linear second order differential equations, including numerical methods such as Euler's method or Runge-Kutta methods, as well as analytical methods such as series solutions or substitution methods. The method used depends on the specific equation and its properties.

## 5. What are the limitations of solving non linear second order differential equations?

Non linear second order differential equations can be very difficult to solve, and in some cases, there may not be an exact solution. In these cases, numerical methods or approximation techniques must be used to find an approximate solution. Additionally, the complexity of these equations can make it difficult to accurately predict the behavior of the system being modeled.

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